diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzeiqz" "b/data_all_eng_slimpj/shuffled/split2/finalzzeiqz"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzeiqz"
@@ -0,0 +1,5 @@
+{"text":"\\subsection*{\\underline{#1}\\nopunct}}\n\\newcommand\\repart[1]{\\mathrm{Re}\\left[ #1 \\right]}\n\\newcommand\\impart[1]{\\mathrm{Im}\\left[ #1 \\right]}\n\\newcommand\\syl[1]{\\| #1 \\|_{\\mathrm{syl}}}\n\\newcommand\\stl[1]{\\| #1 \\|_{\\mathrm{*}}}\n\\newcommand\\ba{\\begin{align*}}\n\\newcommand\\ea{\\end{align*}}\n\\newcommand\\be{\\begin{enumerate}}\n\\newcommand\\ee{\\end{enumerate}}\n\\newcommand\\bp{\\begin{proof}}\n\\newcommand\\ep{\\end{proof}}\n\\newcommand\\bpp{\\begin{prop}}\n\\newcommand\\epp{\\end{prop}}\n\\newcommand\\bpb{\\begin{prob}}\n\\newcommand\\epb{\\end{prob}}\n\\newcommand\\bd{\\begin{defn}}\n\\newcommand\\ed{\\end{defn}}\n\\newcommand\\bh{\\begin{hint}}\n\\newcommand\\eh{\\end{hint}}\n\n\n\\newcommand\\sgn{\\mathrm{sgn}}\n\\newcommand\\stab{\\mathrm{Stab}}\n\\newcommand\\fform[1]{\\langle\\!\\langle #1\\rangle\\!\\rangle}\n\\newcommand\\vform[1]{\\vert #1\\vert}\n\\newcommand\\vstar[1]{\\| #1\\|_{\\mathrm{*}}}\n\\newcommand\\vh[1]{\\| #1\\|_{\\mathcal{H}}}\n\n\\newcommand\\bC{\\mathbb{C}}\n\\newcommand\\bE{\\mathbb{E}}\n\\newcommand\\bN{\\mathbb{N}}\n\\newcommand\\N{\\mathbb{N}}\n\\newcommand\\bR{\\mathbb{R}}\n\\newcommand\\R{\\mathbb{R}}\n\\newcommand\\bQ{\\mathbb{Q}}\n\\newcommand\\Q{\\mathbb{Q}}\n\\newcommand\\bZ{\\mathbb{Z}}\n\\newcommand\\Z{\\mathbb{Z}}\n\\newcommand\\bH{\\mathbb{H}}\n\\renewcommand\\AA{\\mathcal{A}}\n\\newcommand\\BB{\\mathcal{B}}\n\\newcommand\\CC{\\mathcal{C}}\n\\newcommand\\FF{\\mathcal{F}}\n\\newcommand\\GG{\\mathcal{G}}\n\\newcommand\\DD{\\mathcal{D}}\n\\newcommand\\HH{\\mathcal{H}}\n\\newcommand\\KK{\\mathcal{K}}\n\\newcommand\\XX{\\mathcal{X}}\n\\newcommand\\sech{\\operatorname{sech}}\n\\newcommand\\Sym{\\operatorname{Sym}}\n\n\\newcommand\\ev{\\operatorname{ev}}\n\\newcommand\\cay{\\operatorname{Cayley}}\n\\newcommand\\aut{\\operatorname{Aut}}\n\\newcommand\\Inn{\\operatorname{Inn}}\n\\newcommand\\Mono{\\operatorname{Mono}}\n\\newcommand\\PMod{\\operatorname{PMod}}\n\\newcommand\\Hom{\\operatorname{Hom}}\n\\DeclareMathOperator\\Push{\\mathcal{Push}}\n\\DeclareMathOperator\\Forget{\\mathcal{Forget}}\n\\newcommand\\Isom{\\operatorname{Isom}}\n\\newcommand\\UT{\\operatorname{UT}}\n\\newcommand\\Comm{\\operatorname{Comm}}\n\\newcommand\\Symp{\\operatorname{Symp}}\n\\newcommand\\Fill{\\operatorname{Fill}}\n\\newcommand\\supp{\\operatorname{supp}}\n\\newcommand\\Id{\\operatorname{Id}}\n\\newcommand\\lk{\\operatorname{Lk}}\n\\newcommand\\st{\\operatorname{St}}\n\\newcommand\\SL{\\operatorname{SL}}\n\\newcommand\\Diffb{\\operatorname{Diff}_+^{1+\\mathrm{bv}}}\n\\newcommand\\Cb{C^{1+\\mathrm{bv}}}\n\\newcommand\\diam{\\operatorname{diam}}\n\\newcommand\\Gam{\\Gamma}\n\\newcommand\\gam{\\Gamma}\n\\newcommand\\Mod{\\operatorname{Mod}}\n\\newcommand\\PSL{\\operatorname{PSL}}\n\\newcommand\\symp{\\operatorname{Symp}}\n\\DeclareMathOperator\\Homeo{Homeo}\n\\newcommand\\mC{\\mathcal{C}}\n\\newcommand\\mL{\\mathcal{L}}\n\\newcommand\\mM{\\mathcal{M}}\n\\newcommand\\xek{(X^e)_k}\n\\newcommand\\yt{\\widetilde}\n\\newcommand\\sse{\\subseteq}\n\\newcommand\\co{\\colon}\n\\DeclareMathOperator\\tr{tr}\n\\DeclareMathOperator\\Fix{Fix}\n\\DeclareMathOperator\\Out{Out}\n\\DeclareMathOperator\\Aut{Aut}\n\\DeclareMathOperator\\CAT{CAT}\n\\DeclareMathOperator\\Diff{Diff}\n\\DeclareMathOperator\\Exp{Exp}\n\\newcommand\\catz{\\mathrm{CAT}(0)}\n\\DeclareMathOperator\\Stab{Stab}\n\\DeclareMathOperator\\Orbit{Orbit}\n\\newcommand\\var{\\operatorname{var}}\n\\newcommand\\rot{\\operatorname{rot}}\n\\newcommand\\Cbv{C^{1+\\mathrm{bv}}}\n\\DeclareMathOperator\\Per{Per}\n\n\n\n\\renewcommand{\\thefootnote}{\\alpha{footnote}}\n\\newcommand{\\arxiv}[1]\n{\\texttt{\\href{http:\/\/arxiv.org\/abs\/#1}{arXiv:#1}}}\n\\newcommand{\\doi}[1]\n{\\texttt{\\href{http:\/\/dx.doi.org\/#1}{doi:#1}}}\n\\renewcommand{\\MR}[1]\n{\\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{MR#1}}\n\n\n\n\n\n\n\\def{Free products and the algebraic structure of diffeomorphism groups}{{Free products and the algebraic structure of diffeomorphism groups}}\n\\def{Sang-hyun Kim and Thomas Koberda}{{Sang-hyun Kim and Thomas Koberda}}\n\\usepackage{hyperref}\n\\hypersetup{\n  \n   colorlinks=false,\n   plainpages,\n   urlcolor=black,\n   linkcolor=black\n   pdftitle=   {Free products and the algebraic structure of diffeomorphism groups},\n   pdfauthor=  {{Sang-hyun Kim and Thomas Koberda}}\n}\n\n\n\n\\theoremstyle{theorem}\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{lem}[thm]{Lemma}\n\\newtheorem{cor}[thm]{Corollary}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{con}[thm]{Conjecture}\n\\newtheorem{que}[thm]{Question}\n\\newtheorem*{claim*}{Claim}\n\\newtheorem*{fact}{Fact}\n\\newtheorem{claim}{Claim}\n\\newtheorem{thmA}{Theorem}\n\\renewcommand*{\\thethmA}{\\Alph{thmA}}\n\\newtheorem{corA}[thmA]{Corollary}\n\\renewcommand*{\\thecorA}{\\Alph{corA}}\n\n\n\\theoremstyle{remark}\n\\newtheorem{exmp}[thm]{Example}\n\\newtheorem{rem}[thm]{Remark}\n\n\n\\theoremstyle{definition}\n\\newtheorem{defn}[thm]{Definition}\n\\newtheorem{prob}{Problem}[section]\n\\newtheorem{exc}{Exercise}[section]\n\n\\begin{document}\n\\title{Free products and the algebraic structure of diffeomorphism groups}\n\\date{\\today}\n\\keywords{free product; metabelian group; Thompson's group; right-angled Artin group; smoothing; co-graph}\n\\subjclass[2010]{Primary: 57M60; Secondary: 20F36, 37C05, 37C85, 57S05}\n\n\\author[S. Kim]{Sang-hyun Kim}\n\\address{Department of Mathematical Sciences, Seoul National University, Seoul, Korea}\n\\email{s.kim@snu.ac.kr}\n\\urladdr{http:\/\/cayley.kr}\n\n\\author[T. Koberda]{Thomas Koberda}\n\\address{Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA}\n\\email{thomas.koberda@gmail.com}\n\\urladdr{http:\/\/faculty.virginia.edu\/Koberda}\n\n\n\n\n\n\\begin{abstract}\nLet $M$ be a compact one--manifold,\nand let $\\Diffb(M)$ denote the group of $C^1$ orientation preserving diffeomorphisms of $M$ whose first derivatives have bounded variation.\nWe prove that if $G$ is a group which is not virtually metabelian, then $(G\\times\\Z)*\\Z$ is not realized as a subgroup of $\\Diffb(M)$.\nThis  gives the first examples of finitely generated groups $G,H\\le \\Diff_+^\\infty(M)$ such that $G\\ast H$ does not embed into $\\Diffb(M)$.\nBy contrast, \nfor all countable groups $G,H\\le\\Homeo^+(M)$ there exists an embedding $G\\ast H\\to \\Homeo^+(M)$.\nWe deduce that many common groups of homeomorphisms do not embed into $\\Diffb(M)$, for example the free product of $\\bZ$ with Thompson's group $F$.\nWe also complete the classification of right-angled Artin groups which can act smoothly on $M$\nand in particular, recover the main result of a joint work of the authors with Baik~\\cite{BKK2014}. \nNamely, a right-angled Artin group $A(\\gam)$ either admits a faithful $C^{\\infty}$ action on $M$, or $A(\\gam)$ admits no faithful $\\Cb$ action on $M$.  In the former case, $A(\\gam)\\cong\\prod_i G_i$\nwhere $G_i$ is a free product of free abelian groups.\nFinally, we develop a hierarchy of right-angled Artin groups, with the levels of the hierarchy corresponding to the number of semi-conjugacy classes of possible actions of these groups on $S^1$.\n\\end{abstract}\n\n\\maketitle\n\n\n\n\\section{Introduction}\n\nLet $M$ be a compact one--manifold. In this article, we study the algebraic structure of the group $\\Diffb(M)$, where here $\\Diffb(M)$ denotes the group of $C^1$ diffeomorphisms of $M$ whose derivatives have bounded variation.\nSpecifically, we consider the restrictions placed on subgroups of $\\Diffb(M)$ by the $\\Cb$ regularity assumption. Our main result implies that there is a large class $\\AA_0 $ of finitely generated subgroups of $\\Diff_+^\\infty(M)$ such that for all $G,H\\in\\AA_0$, the free product $G\\ast H$ can never be realized as a subgroup of $\\Diffb(M)$; see Corollary~\\ref{cor:classification}.\n\nAs a corollary, we complete a program initiated by Baik and the authors in~\\cite{BKK2014,BKK2016} to decide which right-angled Artin groups admit faithful $C^{\\infty}$ actions on a compact one--manifold, and exhibit many classes of finitely generated subgroups of $\\Homeo^+(M)$ which cannot be realized as subgroups of $\\Diffb(M)$.\n\n\\subsection{Statement of results}\n\nUnless otherwise noted, $M$ will denote a compact one--manifold.\n That is to say, $M$ is a finite union of disjoint closed intervals $I=[0,1]$ and circles $S^1=\\bR\/\\bZ$.\nAn \\emph{action} on a one--manifold is always assumed to mean \nan orientation preserving action. \nFor each $0\\le r\\le\\infty$ or $r=\\omega$, we let $\\Diff^r_+(M)$ denote the group of orientation preserving $C^r$ (analytic if $r=\\omega$) diffeomorphisms.\nWe write $\\Homeo^+(M)=\\Diff^0_+(M)$.\nOur main result is the following:\n\n\\begin{thm}\\label{thm:main}\nIf $G$ is a group which is not virtually metabelian, then the group $(G\\times\\Z)*\\Z$ admits no faithful $C^{1+\\mathrm{bv}}$ action on $M$.\n\\end{thm}\n\n\nTheorem~\\ref{thm:main} clearly implies that $(G\\times\\Z)*\\Z$ does not embed into $\\Diff^r_+(M)$ for every level of regularity $r\\geq 2$.\n A key step in our proof of Theorem~\\ref{thm:main} is the following result on $C^1$--smoothability:\n\\begin{thm}[Theorem~\\ref{t:tech-main}]\\label{thm:tech-main-intro}\nLet $X\\in\\{I,S^1\\}$,\nand let\n $a,b,t\\in\\Diff^1_+(X)$.\n If\n\\[\\supp a\\cap\\supp b=\\varnothing,\\] \nthen the group $\\form{a,b,t}$  is not isomorphic to $\\bZ^2\\ast\\bZ$.\n\\end{thm}\n\n\nWe have stated Theorem~\\ref{thm:main} as above for clarity and concision, though several stronger statements can be deduced from the proof we give. In the particular case of finitely generated groups, we note the following, which should be compared to Corollary~\\ref{cor:cnt}:\n\n\n\\begin{cor}\\label{cor:fg1}\nIf $G$ is a finitely generated group which is not virtually abelian, then $(G\\times\\Z)*\\Z$ admits no faithful $\\Cb$ action on $M$. \n\\end{cor}\nThe motivation for proving Theorem~\\ref{thm:main} came from investigating right-angled Artin subgroups of $\\Diffb(M)$ (cf. Corollary~\\ref{cor:classification} below). The simplest right-angled Artin group to which Theorem~\\ref{thm:main} applies is $(F_2\\times\\Z)*\\Z$:\n\n\n\\begin{cor}\\label{cor:FtimesZ}\nThe group $(F_2\\times\\Z)*\\Z$ is not a subgroup of $\\Diffb(M)$.\n\\end{cor}\n\nSince the hypotheses on Theorem~\\ref{thm:main} are relatively weak, there are many finitely generated subgroups of $\\Homeo^+(M)$ which can be thus shown to admit no faithful $\\Cb$ actions on a compact manifold. Recall that \\emph{Thompson's group $F$} is a group of piecewise linear homeomorphisms of the interval, with dyadic break points and with all slopes being powers of two. \\emph{Thompson's group $T$} is the analogous group defined for the circle. A famous result of Ghys--Sergiescu~\\cite{GS1987} says that the standard actions of $T$ and $F$ are topologically conjugate into a group of $C^{\\infty}$ diffeomorphisms of the circle and of the interval, respectively. However, we have the following:\n\n\\begin{cor}\\label{cor:Thompson}\nThe groups $F*\\Z$ and $T*\\Z$ are not subgroups of $\\Diffb(M)$.\n\\end{cor}\n\nIn the vein of Corollary~\\ref{cor:Thompson}, we do not know the answer to the following:\n\n\\begin{que}\\label{que:thompson-critical}\nAre the groups $F*\\Z$ and $T*\\Z$ subgroups of $\\Diff_+^1(I)$ and $\\Diff_+^1(S^1)$ respectively? If so, what is their optimal regularity, i.e. the supremum of the H\\\"older continuity exponent $\\tau\\in [0,1)$ such that these groups embed in $\\Diff_+^{1+\\tau}$ for the relevant manifolds (cf.~\\cite{JNR2018,KK2017crit})?\n\\end{que}\n\n\n\n\nReturning to the original motivation, Theorem \\ref{thm:main} combined with results of Farb--Franks and Jorquera completes the classification of right-angled Artin groups admitting actions of various regularities on compact one--manifolds. Before stating the result, we define some terminology. We write $\\gam$ for a finite simplicial graph with vertex set $V(\\gam)$ and edge set $E(\\gam)$. The \\emph{right-angled Artin group} (or \\emph{RAAG}, for short) on $\\gam$ is defined as \\[A(\\gam)=\\langle V(\\gam)\\mid [v,w]=1 \\textrm{ if and only if }\\{v,w\\}\\in E(\\gam)\\rangle.\\]\n\nA subgraph $\\Lambda$ of a graph $\\gam$ is called a \\emph{full} subgraph if $\\Lambda$ is spanned by the vertices of $\\Lambda$. That is, two vertices in $\\Lambda$ are adjacent if and only if they are adjacent in $\\gam$.\nA simplicial graph $\\gam$ is called \\emph{$P_4$--free} if no full subgraph of $\\gam$ is isomorphic to a path $P_4$ on four vertices. Such graphs are often called \\emph{cographs}~\\cite{CLB1981,KK2013}. We write $\\mathcal{K}$ for the class of cographs. It is well--known that cographs can be fit into a hierarchy which is defined as follows:\n\\begin{enumerate}\n\\item\nThe class $\\mathcal{K}_0$ consists of a single vertex;\n\\item\nIf $n\\geq 1$ is odd then $\\mathcal{K}_n$ is obtained by taking $\\mathcal{K}_{n-1}$ together with finite joins of elements in $\\mathcal{K}_{n-1}$;\n\\item\nIf $n\\geq 2$ is even then $\\mathcal{K}_n$ is obtained by taking $\\mathcal{K}_{n-1}$ together with finite disjoint unions of elements in $\\mathcal{K}_{n-1}$.\n\\end{enumerate}\nHere, a \\emph{join} of two simplicial graphs $X$ and $Y$ is a simplicial graph consisting of the disjoint union of $X$ and $Y$, together with an edge of the form $\\{x,y\\}$ for every vertex $x$ of $X$ and every vertex $y$ of $Y$.\n\n\nWe have that $\\mathcal{K}_0\\subset\\mathcal{K}_1\\subset\\KK_2\\subset\\cdots$ and \\[\\mathcal{K}=\\bigcup_i\\mathcal{K}_i.\\] \nNote that join and disjoint union correspond to direct product and free product respectively, so that right-angled Artin groups on cographs are exactly the smallest class of groups containing $\\Z$, which is closed under finite direct products, and which is closed under finite free products. The reader will observe that if $\\gam\\in\\mathcal{K}_0$ then $A(\\gam)\\cong\\Z$. Similarly, if $\\gam\\in\\mathcal{K}_1$ then $A(\\gam)$ is free abelian, and if $\\gam\\in\\mathcal{K}_2$ then $A(\\gam)$ is a free product of free abelian groups.\nIf $\\gam\\in\\mathcal{K}_3$ then $A(\\Gamma)$ can be written as \\[A(\\Gamma)=\\prod_{i=1}^m G_i,\\]\n  where each $G_i$ is a free product of free abelian groups.\nIn~\\cite{BKK2016}, Baik and the authors proved that if $A(\\gam)$ admits an injective homomorphism into $\\Diffb(M)$, then $\\gam\\in\\KK$. We will deduce a strengthening of this result, using Theorem~\\ref{thm:main}.\n\n\n\n\n\n\\begin{cor}[cf. \\cite{BKK2016}]\\label{cor:classification}\nLet $A(\\gam)$ be a right-angled Artin group.\n\\begin{enumerate}\n\\item (see \\cite{FF2003,Jorquera})\nThere exists an injective homomorphism $A(\\gam)\\to\\Diff_+^1(M)$; thus, $A(\\gam)$ admits a faithful $C^1$ action of $M$.\n\\item\nIf there exists an injective homomorphism $A(\\gam)\\to\\Diffb(M)$ then $\\gam\\in\\mathcal{K}_3$; conversely, if $\\gam\\in\\mathcal{K}_3$ then $A(\\gam)\\le\\Diff_+^{\\infty}(M)$.\n\\end{enumerate}\n\\end{cor}\n\nIn particular, if we define\n\\[\n\\AA_0=\\{A(\\Gamma)\\mid \\Gamma\\in\\KK_3\\setminus\\KK_2\\}\\]\nthen each $G\\in\\AA_0$ embeds into $\\Diff_+^\\infty(M)$,\nbut for all $G,H\\in\\AA_0$ the group $G\\ast H$ never embeds into $\\Diff_+^{1+\\mathrm{bv}}(M)$.\nBy contrast, we have the following, which is well--known from several contexts.\n\n\\begin{prop}[cf. \\cite{KM1996,Rivas2012JAlg,BS2015}]\\label{prop:free prod}\nThe class of countable subgroups of $\\Homeo^+(M)$ is closed under finite free products.\n\\end{prop}\n\nIn the same spirit of Question~\\ref{que:thompson-critical} and the authors' paper~\\cite{KK2017crit}, we have the following:\n\n\\begin{que}\nLet $\\gam\\notin\\mathcal{K}_3$. What is the supremum of $\\tau\\in [0,1)$ for which $A(\\gam)$ embeds in $\\Diff_+^{1+\\tau}(M)$? Does $\\tau$ depend on $\\gam$?\n\\end{que}\n\nLet $S$ be an orientable surface of genus $g$ and with $n$ punctures or boundary components. We say that $S$ is \\emph{sporadic} if \\[3g-3+n\\leq 1.\\] We write $\\Mod(S)$ for the mapping class group of $S$, i.e. $\\Mod(S)=\\pi_0(\\Homeo^+(S))$. Using the main result of~\\cite{Koberda2012}, we immediately recover the following result as a corollary of Theorem~\\ref{thm:main}:\n\n\\begin{cor}[cf.~\\cite{BKK2016}]\nLet $M$ be a compact one--manifold, and let $S$ be an orientable finite-type surface. Then there exists a finite index subgroup $G\\le\\Mod(S)$ such that $G\\le\\Diffb(M)$ if and only if $S$ is sporadic.\n\\end{cor}\n\n\n\nTheorem \\ref{thm:main} allows us to build a hierarchy on right-angled Artin groups, whose levels correspond to right-angled Artin groups with more or fewer ``dynamically different\" actions on the circle. Roughly speaking, two group actions \\[\\rho_1,\\rho_2\\colon G\\to\\Homeo^+(S^1)\\] are \\emph{semi--conjugate} (or, \\emph{monotone-equivalent}) if there exists \nanother action \\[\\rho\\co G\\to\\Homeo^+(S^1)\\]\nand monotone degree one maps $h_i\\colon S^1\\to S^1$ such that\n\\[h_i\\circ \\rho=\\rho_i\\circ h_i\\] for each $i=1,2$. See~\\cite{Ghys1987,CD2003IM,Ghys2001,Mann-hb,BFH2014,KKMj2016} for instance, and the many references therein.\nA \\emph{projective action} of a group $G$ is a representation\n\\[\n\\rho\\co G\\to \\PSL_2(\\bR),\\]\nwhere $\\PSL_2(\\R)$ sits inside of $\\Homeo^+(S^1)$ as the group of projective analytic diffeomorphisms of $S^1$. \n\n\n\\begin{cor}\\label{cor:conj}\nLet $A(\\gam)$ be a right-angled Artin group.\n\\begin{enumerate}\n\\item\nIf $\\gam\\in\\mathcal{K}_2$, then $A(\\gam)$ admits uncountably many distinct semi--conjugacy classes of faithful orientation preserving projective actions on $S^1$;\n\\item\nIf $\\gam\\in\\mathcal{K}_3\\setminus\\mathcal{K}_2$ then any faithful orientation preserving $\\Cb$ action of $A(\\gam)$ on $S^1$ has a periodic point and no dense orbits and hence admits at most countably many distinct semi--conjugacy classes of $\\Cb$ actions on $S^1$;\n\\item\nIf $\\gam\\notin\\mathcal{K}_3$ then $A(\\gam)$ admits no faithful $\\Cb$ action on $S^1$.\n\\end{enumerate}\n\\end{cor}\n\nIn the case of analytic actions on a compact connected one--manifold $M$, one has the following result of Akhmedov and Cohen:\n\n\\begin{thm}[See~\\cite{AC2015TA}]\nThe right-angled Artin group $A(\\gam)$ embeds into $\\Diff^\\omega(M)$ if and only if $\\Gamma\\in\\KK_2$, i.e. $A(\\gam)$ decomposes as a free product of free abelian groups.\n\\end{thm}\n\n\\subsection{Notes and references}\n\nThis paper\nreveals some of the subtlety of the interplay between algebra and regularity in diffeomorphism groups of one--manifolds. \nOur paper arose during the effort to complete the classification of right-angled Artin subgroups of $\\Diff_+^{\\infty}(S^1)$\nin the spirit of~\\cite{BKK2016}.\nThe essential content of this paper is Lemma~\\ref{l:main}, which exhibits an explicit element of the kernel of any given $\\Cb$ action of $(G\\times\\Z)*\\Z$ action on a compact one--manifold.\nSince a group of the form $(G\\times\\Z)*\\Z$ is generally simpler than the right-angled Artin groups considered in~\\cite{BKK2016}, it is more difficult to find elements in the kernel of a given action, and therefore we develop more sophisticated tools here. We note that our main result does subsume the main result of~\\cite{BKK2016}. Indeed, the main result of~\\cite{BKK2016} is that there is no injective homomorphism $A(P_4)\\to\\Diffb(M)$, where here \\[A(P_4)=\\langle a,b,c,d\\mid [a,b]=[b,c]=[c,d]=1\\rangle.\\] The group $A(P_4)$ contains a copy of $(F_2\\times\\Z)*\\Z$, which cannot embed in $\\Diffb(M)$ by Corollary~\\ref{cor:FtimesZ}. An explicit embedding of $(F_2\\times\\Z)*\\Z$ into $A(P_4)$ is given by $\\langle a,b,c,dad^{-1}\\rangle\\le A(P_4)$ (see~\\cite{KK2013} for a discussion of this fact).\n\nThe program completed by Corollary~\\ref{cor:classification} fully answers a question raised in a paper of M. Kapovich (attributed to Kharlamov) as to which right-angled Artin groups admit faithful $C^{\\infty}$ actions on the circle~\\cite{Kapovich2012}.\n\nRight-angled Artin subgroups of diffeomorphism groups of one--manifolds find an analogue in right-angled Artin subgroups of linear groups. It is well--known that right-angled Artin groups are always linear over $\\Z$ and hence admit injective homomorphisms into $\\SL_n(\\Z)$~\\cite{Humphries1994,HW1999,DJ2000}. For $\\SL_3(\\Z)$, it is still unclear which right-angled Artin groups appear as subgroups. Long--Reid~\\cite{LR2011} showed that $F_2\\times\\Z$ is not a subgroup of $\\SL_3(\\Z)$, which implies that any right-angled Artin subgroup of $\\SL_3(\\Z)$ is a free product of free abelian groups of rank at most two. It is currently unknown whether or not $\\Z^2*\\Z$ is a subgroup of $\\SL_3(\\Z)$.\n\nA consequence of the technical work behind Theorem~\\ref{thm:main} is a certain criterion to prove that a group contains a lamplighter subgroup. See Lemma~\\ref{l:recursive} and Proposition~\\ref{t:tech-main2} for precise statements.\n\nA crucial step in our proof of the main theorem is a $C^1$--rigidity result, which is Theorem~\\ref{t:tech-main}. \nWe note that Thurston~\\cite{Thurston1974Top}, Calegari~\\cite{Calegari2008AGT}, Navas~\\cite{Navas2008GAFA} and Bonatti, Monteverde, Navas and Rivas~\\cite{BMNR2017MZ} explored various remarkable $C^1$--rigidity results. \nIn particular, a $C^1$--rigidity result on Baumslag--Solitar groups in~\\cite{BMNR2017MZ} was employed in a very recent paper by Bonatti, Lodha and Triestino~\\cite{BLT2017},\nto produce certain piecewise affine homeomorphism groups of $\\bR$ which do not embed into $\\Diff^1_+(I)$.\n\nAs for other classes of groups of homeomorphisms which cannot be realized as groups of $\\Cb$ diffeomorphisms, Corollary~\\ref{cor:Thompson} is a complement to a result of the second author with Lodha~\\cite{KL2017}, in which they show that certain ``square roots\" of Thompson's group $F$ may fail to act faithfully by $\\Cb$ diffeomorphisms on a compact one--manifold, even though they are manifestly groups of homeomorphisms of these manifolds. Thus, the Ghys--Sergiescu Theorem appears to place $F$ and $T$ at the cusp of smoothability in the sense that even relatively minor algebraic variations on $F$ and on $T$ fail to be smoothable.\n\nFinally, we remark on the optimality of the differentiability hypothesis. \nCorollary~\\ref{cor:classification} shows that every right-angled Artin group can act faithfully by $C^1$ diffeomorphisms, but nearly none of them can act by $\\Cb$ diffeomorphisms. \nAs for Corollary~\\ref{cor:not closed} and Proposition~\\ref{prop:free prod}, we have the following result (based on~\\cite{BMNR2017MZ} and on a suggestion by Navas), which is proved in the authors' recent manuscript~\\cite{KK2017crit}:\n\n\\begin{prop}[\\cite{KK2017crit}]\\label{prop:c1freeproduct}\nLet $M\\in\\{I,S^1\\}$, and let $BS(1,2)$ be the Baumslag--Solitar group with the presentation $\\langle s,t\\mid sts^{-1}=t^2\\rangle$. Then the group $(BS(1,2)\\times\\Z)*\\Z$ is not a subgroup of $\\Diff_+^1(M)$. In particular, the class of finitely generated subgroups of $\\Diff_+^1(M)$ is not closed under taking finite free products.\n\\end{prop}\n\n\\section{Background on one--dimensional smooth dynamics}\nWe very briefly summarize the necessary background from one--dimensional dynamics. The reader may also consult~\\cite{BKK2016}, parts of which we repeat here, and where we direct the reader for proofs.\n\n\\subsection{Poincar\\'e's theory of rotation numbers}\n\nLet $f\\in\\Homeo^+(S^1)$, and let $\\tilde f\\co \\bR\\to\\bR$ be  an arbitrary lift of $f$.\nThen the \\emph{rotation number} of $f$ is defined as\n\\[\n\\rot f = \\lim_{n\\to\\infty}\\frac{\\tilde f^n(x)}{n}\\in \\bR\/\\bZ=S^1\\]\nwhere $x\\in \\bR$. Then $\\rot f$ is well-defined, and independent of the choice of a lift $\\tilde f$ and a base point $x\\in \\bR$; see~\\cite{Navas2011} for instance. \nThe set of periodic points of $f$ is denoted as $\\Per f$.\nLet us record some elementary facts.\n\\begin{lem}\\label{l:inv}\nFor $f\\in\\Homeo^+(S^1)$, the following hold.\n\\be\n\\item\n$\\rot(f) =0$ if and only if $\\Fix f\\ne\\varnothing$.\n\\item\n$\\rot(f)\\in\\bQ$ if and only if $\\Per f\\ne\\varnothing$.\n\\item\\label{p:inv}\nIf $x\\in S^1$ and $g\\in\\Homeo^+(S^1)$ satisfy\n\\[\nf^n(x)=g^n(x)\\]\nfor all $n\\in\\bZ$, then $\\rot(f)=\\rot(g)$.\n\\ee\n\\end{lem}\n\nThe rotation number is a continuous class function (that is, constant on each conjugacy class)\n\\[\n\\rot\\co \\Homeo^+(S^1)\\to S^1.\\]\nMoreover, the rotation number restricts to a group homomorphism on each amenable subgroup of $\\Homeo^+(S^1)$; see~\\cite{Ghys2001}.\n\nLet us recall the following classical result.\n\n\n\\begin{thm}[H\\\"older's Theorem~\\cite{Holder1901}; see~\\cite{Navas2011}]\\label{t:hoelder}\nA group acting freely on $\\bR$ or on $S^1$ by orientation preserving homeomorphisms is abelian.\n\\end{thm}\n\nWe will use the following variation, which is similar to \\cite[Theorem 2.2]{FF2001}.\n\\begin{cor}[cf.~\\cite{FF2001}]\\label{c:hoelder-s1}\nLet  $X$ be a nonempty closed subset of $S^1$,\nand let $G$ be a group acting freely on $X$  by orientation preserving homeomorphisms.\nThen  the action of $G$ extends to a free action $\\rho\\co G\\to\\Homeo^+(S^1)$ such that \n\\[\\rot\\circ\\rho\\co G\\to S^1\\] is an injective group homomorphism.\n\\end{cor}\n\\bp\nIf $(a,b)$ is a component of $S^1\\setminus X$ and $g \\in  G$, then $(ga,gb)$ is also a component of $S^1\\setminus X$. So, $G$ extends to some action $\\rho$  on $S^1$ in an affine manner.  \nPut \\[S^1\\setminus X = \\coprod_{i\\ge1} I_i.\\] \nIf $\\rho (g)y = y$ for some $g \\in  G$ and for some $y \\in  S^1$, then $y \\in  I_i$ for some $i$. We have that $ \\rho (g)$ restricts to the identity on $ I_i$ by definition. This would imply $\\rho (g)\\partial I_i = g \\partial I_i = \\partial I_i$, and so, $g = 1$. That is, the action $\\rho$  of $G$ on $S^1$ is free.\n\nBy H\\\"older's theorem, we see $\\rho(G)\\cong G$ is abelian. Since abelian groups are amenable, \nwe have that $\\rot \\circ \\rho$ is a group homomorphism. The freeness of the action $\\rho$ implies that $\\rot\\circ\\rho(g)\\ne0$ for all nontrivial $g$.\\ep\n\n\n\n\\subsection{Kopell--Denjoy theory}\nLet $M\\in\\{I,S^1\\}$.\nWe denote by $\\var(g;M)$ the total variation of a map $g\\co M\\to\\bR$:\n\\[\\var(g;M) = \\sup\\left\\{\\sum_{i=0}^{n-1}\\left| g(a_{i+1})- g(a_i)\\right|\n\\co (a_i\\co 0\\le i\\le n)\\text{ is a partition of }M\\right\\}.\\]\nIn the case $M=S^1$, we require $a_n=a_0$ in the above definition.\nFollowing \\cite{Navas2011},\nwe say a $C^1$ diffeomorphism $f$ on $M$ is $\\Cbv$ if $\\var(f';M)<\\infty$.\nWe let $\\Diffb(M)$ denote\nthe group of orientation preserving $\\Cbv$ diffeomorphisms of $M$.\nThe following two results play a fundamental role on the study of $\\Cbv$ diffeomorphisms.\n\n\\begin{thm}[Denjoy's Theorem {\\cite{Denjoy1958}, \\cite{Navas2011}}]\\label{t:denjoy}\nIf $a\\in\\Diffb(S^1)$ and $\\Per a=\\emptyset$, then $a$ is topologically conjugate to an irrational rotation.\n\\end{thm}\n\n\\begin{thm}[Kopell's Lemma {\\cite{Kopell1970}, \\cite{Navas2011}}]\\label{t:kopell}\nSuppose $a\\in\\Diffb[0,1)$, $b\\in\\Diff_+^1[0,1)$, and $[a,b]=1$.\nIf $\\Fix a\\cap (0,1)=\\varnothing$ and $b\\ne 1$,\nthen $\\Fix b\\cap (0,1)=\\varnothing$.\n\\end{thm}\nWe remark that the original statement by Kopell was for $C^2$--regularity. Navas extended her result to $C^{1+\\mathrm{bv}}$ case~\\cite[Theorem 4.1.1]{Navas2011}.\n\nLet $X$ be a topological space. Then we define the \\emph{support} of $h\\in \\Homeo(X)$ as\n\\[\\supp h = X\\setminus\\Fix h.\\] \nIt is convenient for us to consider the non--standard \\emph{open support} of a homeomorphism as defined here, whereas many other authors use the closure of the open support. We will consistently mean the  open support unless otherwise noted.\n\nFor a subgroup $G\\le\\Homeo(X)$,  we put\n\\[\\supp G = \\bigcup_{g\\in G}\\supp g.\\]\nWe say that $f\\in\\Homeo(X)$ is \\emph{grounded} if $\\Fix f\\neq\\varnothing$. We note that every $f\\in\\Homeo^+(I)$ is grounded by definition.\n\nThe following important observations on commuting $\\Cbv$ diffeomorphisms essentially builds on Kopell's Lemma and H\\\"older's Theorem. \n\n\n\\begin{lem}\\label{l:disj-abel}\nThe following hold:\n\\be\n\\item (Disjointness Condition,~\\cite{BKK2016})\nLet $M\\in\\{I,S^1\\}$,\nand let $a,b\\in\\Diffb(M)$ be commuting grounded diffeomorphisms.\nIf $A$ and $B$ are components of $\\supp a$ and $\\supp b$ respectively,\nthen either $A=B$ or $A\\cap B=\\varnothing$.\n\\item (Abelian Criterion, cf.~\\cite{FF2001})\nIf $a,b,c\\in\\Diffb(I)$ satisfy $\\Fix a =\\partial I$ and that\n$[a,b]=1=[a,c]$, then $[b,c]=1$.\n\\ee\n\\end{lem}\n\n\\begin{rem}\nThe abelian criterion as given by Farb and Franks is a straightforward consequence of H\\\"older's Theorem and Kopell's Lemma. We thank one of the referees for pointing out  another simple proof using Szekeres' Theorem~\\cite{Navas2011}.\n\\end{rem}\n\nThe notation $f\\restriction_A$ for a function $f$ (or set of functions) and a set $A$ means the restriction to $A$.\nWe will need the following properties of centralizer groups.\n\n\n\\begin{lem}\\label{l:circle}\nLet $a\\in \\Diffb(S^1)$ be an infinite order element,\nand let $Z(a)$ be the centralizer of $a$ in $\\Diffb(S^1)$.\nThen the following hold.\n\\be\n\\item\\label{p:zz}\nIf $a$ is grounded and if a group $H\\le Z(a)$ is generated by grounded elements, then every element in $H$ is grounded and moreover,\n\\[\\supp a\\cap \\supp [H,H]=\\varnothing.\\]\n\\item\\label{p:rot-irr}\nIf $\\rot a\\not\\in\\bQ$, then $Z(a)$ is topologically conjugate to a subgroup of $\\operatorname{SO}(2,\\bR)$.\n\\item\\label{p:rot-q}\nIf $\\rot a\\in\\bQ$, then $\\rot Z(a)\\sse\\bQ$.\n\\item\\label{p:hom} \n({cf. \\cite[Lemma 3.4]{FF2001}})\nThe rotation number restricts to a homomorphism on $Z(a)$;\nin particular, every element of $[Z(a),Z(a)]$ is grounded.\n\\ee\n\\end{lem}\n\\bp\n(\\ref{p:zz})\nLet $J$ be a component of $\\supp a$.\nBy the Disjointness Condition, the group $H$ acts on the open interval $J$. Since $H$ fixes $\\partial J$, every element of $H$ is grounded.\nThe Abelian Criterion implies that\n\\[[H,H]\\restriction_J=1.\\]\nSo, we have that $J\\cap \\supp[H,H]=\\varnothing$.\n\n(\\ref{p:rot-irr}) By Denjoy's Theorem, the map $a$ is topologically conjugate to an irrational rotation. The centralizer of an irrational rotation in $\\Homeo^+(S^1)$ is $\\operatorname{SO}(2,\\bR)$;\nsee~\\cite[Proposition 2.10]{FF2001} or~\\cite[Exercise 2.2.12]{Navas2011}. \n\n\n\n(\\ref{p:rot-q})\nSuppose some element $b$ in $Z(a)$ has an irrational rotation number. \nSince $a\\in Z(b)$, part (\\ref{p:rot-irr}) implies that $a$ is conjugate to a rotation. This is a contradiction, for a rotation with a rational rotation number must have a finite order.\n\n(\\ref{p:hom})\nIf $\\rot a$ is irrational, then the conclusion follows from part (\\ref{p:rot-irr}). So we may assume $\\rot a\\in\\bQ$. Then $a^p$ is grounded for some $p\\ne0$.\nSince $Z(a)\\le Z(a^p)$, it suffices to prove the lemma for $a^p$. In other words, we may further suppose that $a$ is grounded.\n\nLet us put \n\\[G = Z(a),\\quad G_0 = \\rot^{-1}(0)\\cap G.\\]\nPart (\\ref{p:zz}) implies that \n\\[\nG_0=\\bigcup_{x\\in S^1}\\stab_G(x)=\\form{G_0}\\]\nis a group.\nSince $\\rot$ is a class function on $\\Homeo^+(S^1)$,\nwe see that $G_0\\unlhd G$.\n\nFor each $x\\in \\Fix a$ and $g\\in G$, we note\n\\[\nag(x) = ga(x) = g(x).\\]\nSo, $\\Fix a$ is $G$--invariant. \nWe have a nonempty proper closed $G$--invariant set\n\\[\nX=\\partial \\Fix a.\\]\n\\begin{claim}\\label{claim:HG}\nFor all $x\\in X$, the group $G_0$ fixes $x$.\n\\end{claim}\nFor each $J\\in\\pi_0\\supp a$ and $g\\in G_0$, \nwe have seen in part (\\ref{p:zz}) that $\\partial J\\sse \\Fix g$.\nSince we can write\n\\[\nX = \\overline{\\bigcup\\{\\partial J\\mid J\\in\\pi_0\\supp a\\}}\\]\nwe see that $X\\sse \\Fix g$. This proves the claim.\n\nLet $p\\co G\\to G\/G_0$ denote the quotient map.\nBy Claim~\\ref{claim:HG}, \n the natural action\n\\[G\/G_0\\to \\Homeo^+(X),\\quad p(g).x=g(x)\\] \nis well-defined and free. \nBy Corollary~\\ref{c:hoelder-s1}, this free action extends to a free action \\[\\rho\\co G\/G_0\\to \\Homeo^+(S^1)\\] such that $\\rot\\circ\\rho$ is an injective homomorphism.\n\nFor each $g\\in G$, $x\\in X$ and $n\\in\\bZ$, we have\n\\[\\rho\\circ p(g^n)(x)=g^n(x).\\]\nBy Lemma~\\ref{l:inv} (\\ref{p:inv}) we see that\n\\[\\rot\\circ\\rho\\circ p = \\rot\\restriction_G,\\]\nand hence, that $\\rot\\restriction_G$ is a group homomorphism.\nAs $S^1$ is abelian, we also obtain \n\\[\n[G,G]\\le \\ker(\\rot\\restriction_G)=G_0.\\qedhere\\]\n\\ep\n\n\n\nNote that a finitely generated subgroup of $\\operatorname{SO}(2,\\bR)$ consisting of elements with rational rotation numbers is necessarily finite. So in Lemma~\\ref{l:circle} (\\ref{p:rot-q}), if $G$ is a finitely generated subgroup of $Z(a)$ then $\\rot(G)$ is a finite subgroup of $S^1\\cong\\operatorname{SO}(2,\\bR)$.\n\n\n\n\n\\begin{rem}\nPart (\\ref{p:hom}) of Lemma~\\ref{l:circle} appears in the unpublished work of Farb and Franks \\cite{FF2001}, on which our argument is based. We included here a detailed, self-contained proof for readers' convenience. We also remark the necessity of the infinite--order hypothesis, which was omitted in~\\cite{FF2001}. For example, let us consider $a,b,c\\in\\Diff^\\infty_+(S^1)$ such that \nfor each $x\\in S^1=\\bR\/\\bZ$ \nwe have\n\\[\na(x)=x+1\/2,\\quad\nb(x)=x+1\/4,\\quad\nc(x+1\/2)=c(x)+1\/2\\]\nand such that\n\\[\nc(0) = 0,\\quad\nc(1\/8) = 1\/4,\\quad\nc(1\/4) = 3\/8.\\]\nThen $b,c\\in Z(a)$ and $(bc)^3(0)=0$. Hence we have\n\\[\\rot (b)+\\rot(c) = 1\/4 + 0 \\ne 1\/3 =\\rot(bc).\\]\n\\end{rem}\n\n\n\\begin{lem}\\label{l:interval}\nLet $a\\in \\Diffb(I)$,\nand let $Z(a)$\nbe the centralizer of $a$ in $\\Diffb(I)$.\nThen we have\n\\[\\supp a\\cap \\supp [Z(a),Z(a)]=\\varnothing.\\]\n\\end{lem}\nThe proof is almost identical to that of Lemma~\\ref{l:circle} (\\ref{p:zz}).\n\n\n\n\n\\subsection{The Two--jumps Lemma}\nThe Two--jumps Lemma was developed by Baik and the authors in~\\cite{BKK2016} and is the second essential analytic result needed to establish Theorem~\\ref{thm:main}.\n\n\n\\begin{lem}[Two--jumps Lemma,~\\cite{BKK2016}]\\label{l:fg}\nLet $M\\in\\{I,S^1\\}$ and let $f,g\\co M\\to M$ be continuous maps.\nSuppose  $(s_i), (t_i)$ and $(y_i)$ are infinite sequences of points in $M$\nsuch that for each $i\\ge1$, one of the following two conditions hold:\n\\be[(i)]\n\\item\n$f(y_i)\\le s_i = g(s_i) < y_i < t_i = f(t_i) \\le g(y_i)$;\n\\item\n$g(y_i)\\le t_i = f(t_i) < y_i < s_i = g(s_i) \\le f(y_i)$.\n\\ee\nIf $|g(y_i)-f(y_i)|$ converges to $0$ as $i$ goes to infinity,\nthen $f$ or $g$ fails to be $C^1$.\n\\end{lem}\n\nFigure~\\ref{f:fg} illustrates the case (i) of Lemma~\\ref{l:fg}.\nThe reader may note that the homeomorphisms $f$ and $g$ above are \\emph{crossed elements}~\\cite[Definition 2.2.43]{Navas2011}.\nIndeed,  the Two--jumps Lemma generalizes a unpublished lemma of Bonatti--Crovisier--Wilkinson regarding crossed $C^1$--diffeomorphisms,\nwhich can be found in \\cite[Proposition 4.2.25]{Navas2011}.\n\n\\begin{figure}[h!]\n  \\tikzstyle {bv}=[black,draw,shape=circle,fill=black,inner sep=1pt]\n\\begin{tikzpicture}[>=stealth',auto,node distance=3cm, thick]\n\\draw  (-4,0) node (1) [bv] {} node [below]  {\\small $f(y_i)$} \n-- (-2,0)  node  (2) [bv] {} node [below] {\\small $s_i$}\n-- (0,0) node (3) [bv] {} node [below] {\\small $y_i$} \n-- (2,0)  node (4) [bv] {} node [below] {\\small $t_i$}\n-- (4,0) node (5) [bv] {} node [below]  {\\small $g(y_i)$};\n\\path (3) edge [->,bend right,red] node  {} (1);\n\\path (3) edge [->>,bend right,blue] node  {} (5);\n\\draw [->>, blue] (2)  edge [out = 200,in=-20,looseness=50] (2);\n\\draw [->, red] (4)  edge [out = 20,in=160,looseness=50] (4);\n\\end{tikzpicture}%\n\\caption{Two--jumps Lemma.}\n\\label{f:fg}\n\\end{figure}\n\n\n\n\n\\section{The $C^1$--Smoothability of $\\bZ^2\\ast\\bZ$}\nThis section establishes the main technical result of the paper as below.\n\n\n\\begin{thm}\\label{t:tech-main}\nLet $M\\in\\{I,S^1\\}$,\nand let\n $a,b,t\\in\\Diff^1_+(M)$. If \n\\[\\supp a\\cap\\supp b=\\varnothing,\\] \nthen the group $\\form{a,b,t}$  is not isomorphic to $\\bZ^2\\ast\\bZ$.\n\\end{thm}\n\n\n\\begin{rem}\n\\be\n\\item\nWe emphasize that this theorem is about $C^1$, rather than $\\Cbv$, diffeomorphisms. \n\\item The regularity hypothesis of $C^1$ cannot be replaced by $C^0$; see Proposition~\\ref{prop:ab-homeo}.\n\\item The proof of Theorem~\\ref{t:tech-main} is relatively easy and standard if  $\\supp a$ or $\\supp b$ is assumed to have finitely many components.\n\\ee\n\\end{rem}\n\nLet us prove Theorem~\\ref{t:tech-main} through a sequence of lemmas in this section.\n\n\\subsection{Finding a lamplighter group from compact support}\nA crucial step in the proof of Theorem~\\ref{t:tech-main} is the following construction,\nwhich generalizes a result of Brin and Squier in the PL setting~\\cite{BS1985}.\nThe same idea to find vanishing words from successive commutators goes back even to the Zassenhaus Lemma~\\cite{Raghunathan1972};\nthe authors thank an anonymous referee for suggesting us to further prove the existence of a lamplighter subgroup.\n\n\\begin{lem}\\label{l:recursive}\nLet $1\\ne g\\in H\\le\\Homeo^+(I)$. If the closure of $\\supp g$ is contained in $\\supp H$, then $H$ contains the lamplighter group $\\bZ\\wr\\bZ$.\n\\end{lem}\n\nMore precisely, we will show that for $g_1=g$, there exists a positive integer $m$ and elements $u_1,\\ldots,u_m\\in H$ such that the recursively defined sequence \\[ g_{i+1}=[g_i,u_i g_i u_i^{-1}],\\quad i=1,2,\\ldots,m,\\] satisfies that\n$\\form{g_m,u_m}\\cong\\bZ\\wr\\bZ$\nand that $g_{m+1}=1$.\nHere and throughout this paper, when $1$ refers to a homeomorphism or a group element then it means the identity, and otherwise it refers to the real number $1$.\n\n\n\n\\bp[Proof of Lemma~\\ref{l:recursive}]\nSince $\\overline{\\supp g_1}$ is a compact subset of the open set $\\supp H$, we can enumerate\n\n\\[ I_1,I_2,\\ldots,I_N\\in\\pi_0(\\supp H)\\]\nsuch that $\\supp g_1\\cap I_i\\ne\\varnothing$ for each $i$ and such that\n\n \\[\\overline{\\supp g_1}\\sse \\bigcup_{i=1}^N I_i.\\]\nNote that each $I_i$ is an open, $H$--invariant interval contained in $(0,1)$.\n\nLet us inductively construct the elements $u_1,\\ldots,u_{k-1}$ satisfying the required properties.\nAs a base case, we put $r(1)=1\\in\\bN$ and \n\\[ K_1 :=\\overline{\\supp g_1}\\cap I_1.\\]\nSince $K_1$ is a nonempty compact subset of $I_1\\sse\\supp H$, we have that\n\\[\n\\sup\\{ u(\\inf K_1)\\mid u\\in H\\}=\\sup I_1>\\sup K_1.\\]\nSo, there exists $u_1\\in H$ such that $K_1\\cap u_1^j K_1=\\varnothing$ for all $j\\in\\bN$. Note that \n\\[\\form{g_1,u_1}\\restriction_{I_1}\\cong\\form{s,t\\mid \\left[s,t^j st^{-j}\\right]=1\\text{ for all }j\\in\\bN}=\\bZ\\wr\\bZ.\\]\nLet us set \\[ r(2) = 1+\\sup\\{ s\\in [1,N]\\mid \\supp [g_1, u_1^j g_1 u_1^{-j}]\\restriction_{I_s}=1\\text{ for all }j\\in\\bN\\}\\ge 1+r(1)=2.\\]\nIf $r(2)>N$, then we have a sequence of surjections\n\\[\n\\bZ\\wr\\bZ\\twoheadrightarrow\n\\form{g_1,u_1}\\twoheadrightarrow\n\\form{g_1,u_1}\\restriction_{I_1}\\twoheadrightarrow\\bZ\\wr\\bZ,\n\\]\nwhich composes to the identity. In particular, $\\form{g_1,u_1}\\cong\\bZ\\wr\\bZ$.\nIn the case where $r(2)\\le N$, we pick $j\\in\\bN$ such that the element\n\\[g_2:= \\left[ g_1, u_1^j g_1 u^{-j}\\right]\\]\nsatisfies $\\supp g_2\\cap I_{r(2)}\\ne\\varnothing$,\nand apply the same argument to $g_2$.\n\nBy a straightforward induction, we eventually find $1\\le m\\le r(m)\\le N$\nand $g_m, u_m\\in G$ such that the following hold:\n\\begin{align*}\n&\\overline{\\supp g_m} \\sse I_{r(m)}\\cup\\cdots\\cup I_N,\\\\\n&\\supp g_m\\cap I_{r(m)}\\ne\\varnothing,\\\\\n&\\supp g_m\\cap u_m^j \\supp g_m\\cap I_{r(m)}=\\varnothing,\\quad\\text{ for all }j\\in\\bN,\\\\\n&\\left[g_m, u_m^j g_m u_m^{-j}\\right] =1\\quad\\text{ for all }j\\in\\bN.\n\\end{align*}\nIt follows that $\\form{g_m,u_m}\\cong \\bZ\\wr\\bZ$.\n\\ep\n\n\n\nLemma~\\ref{l:recursive} implies the following for circle homeomorphisms.\n\n\\begin{lem}\\label{l:global}\nLet $a,b,c,d\\in\\Homeo^+(S^1)$ be nontrivial elements such that\n\\[\n\\supp a \\cap \\supp b = \\varnothing\\quad\\text{ and }\\quad\n\\supp c \\cap \\supp d = \\varnothing.\\]\nIf $\\supp G= S^1$, then $G$ contains $\\bZ\\wr\\bZ$.\n\\end{lem}\n\n\\bp\nFor simplicity, let us abbreviate\n$\\AA  = \\pi_0\\supp a$, and similarly define \n$\\BB$,  $\\CC$ and $\\DD$.\nSince $S^1$ is compact, there exists a finite open covering $\\mathcal{V}$ of $S^1$ such that \n\\[\n\\mathcal{V}\\sse\\AA\\cup \\BB\\cup\\CC\\cup\\DD.\\]\n\nBy minimizing the cardinality,  we can require that $\\mathcal{V}$ forms a \\emph{chain of intervals}.\nMore precisely, this means that\n$\\mathcal{V}=\\{V_1,\\ldots,V_k\\}$ for some $k\\ge1$\nand that \\[\\inf V_i<\\sup V_{i-1}\\le\\inf V_{i+1}\\]\nfor $i=1,2,\\ldots,k$, where here the indices are taken cyclically.\n\nWithout loss of generality, let us assume $V_1\\in\\AA $.\nThen we have $V_{2i-1}\\in\\AA \\cup \\BB$\nand $V_{2i}\\in\\CC\\cup\\DD$ for each $i$. Note that $k$ is an even number\nand that $x=\\inf V_2$ is a global fixed point of $H=\\form{b,c,d}$.\nIn particular, we can regard $H$ as acting on $I$, which is a two-point compactification of $S^1\\setminus \\{x\\}$.\nNote that\n\\[\n\\varnothing\\ne \\overline{\\supp b}\\sse S^1\n\\setminus \\bigcup(\\mathcal{V}\\cap\\AA ) \\sse \\bigcup(\\mathcal{V}\\cap(\\BB\\cup\\CC\\cup\\DD))\n\\sse\\supp H.\\]\nThe desired conclusion follows from Lemma~\\ref{l:recursive}.\n\\ep\n\n\n\\subsection{Supports of commutators}\nWe will need rather technical estimates of supports as given in this subsection.\nIn order to prevent obfuscation of the ideas, we have included some intuition behind the proofs when appropriate. \n\\begin{lem}\\label{l:comm-supp}\nIf $f$ and $g$ are homeomorphisms of a topological space $X$, then\n\\[\\overline{\\supp[f,g]}\\sse \\supp f \\cup \\supp g \\cup \\overline{\\supp f\\cap\\supp g}.\\]\n\\end{lem}\n\\bp\nSuppose \n\\[x\\not\\in \\supp f\\cup \\supp g\\cup \\overline{\\supp f\\cap \\supp g}.\\]\nThen $f(x)=x=g(x)$. Moreover, for some open neighborhood $U$ of $x$ we have \n\\[U\\cap \\supp f\\cap \\supp g =\\varnothing.\\]\nWe can find an open neighborhood $V\\sse U$ of $x$ such that\n\\[f^{\\pm1}(V)\\cup g^{\\pm1}(V)\\sse U.\\]\nLet $y\\in V$. We see $[f,g](y)=y$, by considering the following three cases separately:\n\\[\ny\\in V\\cap \\supp f,\\quad y\\in V\\cap \\supp g,\\quad y\\in V\\cap\\Fix f\\cap \\Fix g.\\]\nSo we obtain that\n\\[[f,g]\\restriction_V=1.\\]\nThis implies  \\[x\\not\\in\\overline{\\supp [f,g]}.\\qedhere\\]\n\\ep\n\n\\begin{lem}\\label{lem:c0} \nLet $X$ be a topological space.\nIf $b,c,d\\in\\Homeo(X)$ satisfy \\[\\supp c\\cap\\supp d= \\varnothing,\\]\nthen for $\\phi=[c,bdb^{-1}]$ we have that\n\\[\n\\supp\\phi \\sse \\supp b \\cup cb(\\supp b \\cap\\supp d)\\cup db^{-1}(\\supp b \\cap \\supp c).\\]\n\\end{lem}\n\nLet us briefly explain the key idea behind the statement of this lemma. The support of the homeomorphism $bdb^{-1}$ is exactly $b(\\supp d)$. The homeomorphism $\\phi$ may be viewed as a composition of $bd^{-1}b^{-1}$ and the conjugate of $bdb^{-1}$ by $c$, and the latter of these has support $cb(\\supp d)$. \nBy exhaustively checking the possible images of points under $bdb^{-1}$ and $cbdb^{-1}c^{-1}$, we see that every $x\\in\\supp \\phi$ belongs to one of the three sets as stated in the lemma.\n\n\\bp[Proof of Lemma~\\ref{lem:c0}]\nFor brevity, let us write \n\\[\n\\tilde b = \\supp b,\\quad\n\\tilde c = \\supp c,\\quad\n\\tilde d = \\supp d.\\]\nLet us consider three equivalent expressions for $\\phi$:\n\\[\n [c,bdb^{-1}]=cb d (cb)^{-1}\\cdot bd^{-1}b^{-1}= c\\cdot b\\cdot db^{-1}c^{-1}(db^{-1})^{-1}\\cdot b^{-1}.\\]\nAfter some set theoretic computation, one sees the following.\n\\begin{align*}\\supp  \\phi&\\sse\\left(\\tilde c\\cup b\\tilde d\\right)\\cap \\left(cb \\tilde d\\cup b\\tilde d\\right)\\cap\\left(\\tilde c\\cup \\tilde b\\cup db^{-1}\\tilde c\\right)\\\\&\\sse\\left((\\tilde c\\cap cb \\tilde d)\\cup b\\tilde d\\right)\\cap\\left(\\tilde c\\cup \\tilde b\\cup db^{-1}\\tilde c\\right)\\\\&\\sse(\\tilde c\\cap cb \\tilde d) \\cup \\left( b\\tilde d\\cap (\\tilde c\\cup \\tilde b\\cup db^{-1}\\tilde c)\\right)\\\\&\\sse\\left(\\tilde c\\cap cb \\tilde d\\right) \\cup \\left( (\\tilde b\\cup \\tilde d)\\cap (\\tilde b\\cup\\tilde c\\cup db^{-1}\\tilde c) \\right)\\\\ &\\sse\\left(\\tilde c\\cap cb \\tilde d\\right)\\cup\\tilde b\\cup \\left(\\tilde d\\cap (\\tilde c\\cup db^{-1}\\tilde c)\\right)\\\\ &\\sse\\left(\\tilde c\\cap cb \\tilde d\\right)\\cup\\tilde b \\cup\\left(\\tilde d\\cap db^{-1}\\tilde c\\right). \\end{align*}\nNote that we used $b\\tilde d\\sse \\tilde b\\cup \\tilde d$, and also $\\tilde c\\cap\\tilde d=\\varnothing$.\nIt now suffices for us to prove the following claim:\n\\begin{claim*}\\label{claim:cbbd}\nWe have the following:\n\\begin{align*}\n\\tilde c\\cap cb\\tilde d&\\sse cb\\left(\\tilde b\\cap\\tilde d\\right),\\\\\n\\tilde d\\cap db^{-1}\\tilde c&\\sse db^{-1}\\left(\\tilde b\\cap\\tilde c\\right).\n\\end{align*}\n\\end{claim*}\nTo see the first part of the claim, let us consider \n$x\\in X$ satisfying\n\\[cb(x)\\in \\tilde c\\cap cb \\tilde d.\\]\nThen we have $x\\in \\tilde d$ and $cb(x)\\in \\tilde c$.\nSince\n $\\tilde c\\cap \\tilde d=\\varnothing$\nand $b(x)\\in c^{-1}\\tilde c=\\tilde c$, we see $x\\ne b(x)$.\nIn particular, we have $x\\in\\tilde b$\n and $cb(x)\\in cb(\\tilde b\\cap \\tilde d)$. This proves the first part of the claim. The second part follows by symmetry.\\ep\n \n\\begin{lem}\\label{lem:c1} \nIf $b,c,d\\in\\Diff^1_+(I)$ are given such that \\[\\supp c\\cap\\supp d= \\varnothing,\\]\nthen for $\\phi=[c,bdb^{-1}]$ we have that\n\\[\\overline{\\supp\\phi\\setminus \\supp b}\\sse\\supp c\\cup\\supp d.\\]\n\\end{lem}\n\\bp\nAs in the proof of Lemma~\\ref{l:global}, we let $\\BB=\\pi_0\\supp b$ and $\\CC=\\pi_0\\supp c$.\nLet\n\\[J_B=B\\cup cb(B\\cap\\supp d)\\cup db^{-1}(B\\cap \\supp c)\\]\nfor each $B\\in\\BB$.\nBy Lemma~\\ref{lem:c0}, we have that\n\\[\\supp\\phi\\sse \\bigcup\\{J_B\\mid B\\in\\BB\\}\n=  \\bigcup\\{J_B\\setminus B\\mid B\\in \\BB\\}\\cup \\supp b.\\]\nMoreover, for each $B\\in\\BB$ we note that\n\\[\n\\overline{\nJ_B\\setminus B}\n\\sse \\overline{c(B)\\setminus B}\\cup \\overline{d(B)\\setminus B}\n\\sse\\supp c \\cup\\supp d.\\]\n\n\\begin{claim*}\\label{claim:jbb}\nThe following set is a finite collection of intervals:\n\\[ \\BB_0=\\{B\\in\\BB\\mid J_B\\ne B\\}.\\]\n\\end{claim*}\nWe will employ the $C^1$--hypothesis for this claim. \nLet us write\n\\[\\BB_1=\\{B\\in\\BB\\mid cb(B\\cap \\supp d)\\setminus B\\ne\\varnothing\\},\\ \n\\BB_2=\\{B\\in\\BB\\mid db^{-1}(B\\cap \\supp c)\\setminus B\\ne\\varnothing\\}.\\]\nAssume for a contradiction that $\\BB_0=\\BB_1\\cup\\BB_2$ is infinite. \nWe may suppose $\\BB_1$ is infinite, as the proof is similar when $\\BB_2$ is infinite.\nThere are infinitely many distinct $B_1,B_2,\\ldots\\in \\BB_1$ \nand $x_i\\in B_i\\cap\\supp d$ such that $cb (x_i)\\not\\in B_i$. \nThen we have $C_1,C_2,\\ldots\\in \\CC$ such that\n$b (x_i), cb(x_i)\\in C_i$.\nSince $x_i\\in\\supp d$, we have $x_i\\not\\in C_i$; see Figure~\\ref{f:JB}.\n\nLet us consider the interval $J_i = [x_i,cb (x_i)]$ which contains $b(x_i)$ in the interior, up to switching the endpoints of this interval.\nThen we have\n\\[(b(x_i),cb (x_i)]\\cap \\partial B_i\\ne\\varnothing,\\quad [x_i,b (x_i))\\cap\\partial C_i\\ne\\varnothing.\\]\nWe now apply the Two--jumps Lemma (Lemma~\\ref{l:fg}) to the following parameters\n\\[f=b^{-1},\\ g=c,\\ s_i=\\partial C_i\\cap B_i,\\  t_i=\\partial B_i\\cap C_i,\\ y_i=b(x_i).\\]\nWe deduce that $b$ or $c$ is not $C^1$.\nThis is a contradiction and the claim is proved.\n\nFrom the claim above, we deduce the conclusion as follows.\n\\[\n\\overline{\\supp\\phi\\setminus\\supp b}\n\\sse\\overline{\\bigcup\\{J_B\\setminus B\\mid B\\in\\BB_0\\}}=\\bigcup\\{\\overline{J_B\\setminus B}\\mid B\\in\\BB_0\\}\n\\sse\\supp c\\cup\\supp d.\\qedhere\\]\n\\ep\n\n\\begin{figure}[h!]\n  \\tikzstyle {a}=[black,postaction=decorate,decoration={%\n    markings,%\n    mark=at position 1 with {\\arrow[black]{stealth};}    }]\n  \\tikzstyle {bv}=[black,draw,shape=circle,fill=black,inner sep=1.5pt]\n{\n\\begin{tikzpicture}[thick,scale=.7]\n\\draw [red,ultra thick] (-4,0.5) -- (2,0.5);\n\\draw [blue,ultra thick] (0,0) -- (6,0);\n\\draw [teal,ultra thick] (4,.5) -- (10,.5);\n\\draw (-3,0) node [] {\\small $\\supp d$}; \n\\draw (9,0) node [] {\\small $C_i$}; \n\\draw (3,-.5) node [] {\\small $B_i$}; \n\\draw [dashed] (1,1.4) -- (1,-.3) node [below]  {\\small $x_i$};\n\\draw [dashed] (5,1.4) -- (5,-.3) node [below]  {\\small $b(x_i)$};\n\\draw [dashed] (7.5,1.4) -- (7.5,-.3) node [below]  {\\small $cb(x_i)$};\n\\draw (1,1) -- (7.5,1);\n\\draw (3,1.4) node [] {\\small $J_i$}; \n\\end{tikzpicture}%\n}\n\\caption{Lemma~\\ref{lem:c1}.}\n\\label{f:JB}\n\\end{figure}\n\n\\subsection{Finding compact supports}\nWe will deduce Theorem~\\ref{t:tech-main} from the following, seemingly weaker result.\n\\begin{lem}\\label{l:main}\nLet $M\\in\\{I,S^1\\}$ and let\n$a,b,c,d\\in\\Diff^1_+(M)$.\nIf \n\\[\n\\supp a\\cap\\supp b=\\varnothing,\\quad\n\\supp c\\cap\\supp d=\\varnothing,\\]\nthen the group $\\form{a,b,c,d}$ is not isomorphic to $\\bZ^2\\ast\\bZ^2$.\n\\end{lem}\n\nLet us note two properties of RAAGs. First, a RAAG does not contain a subgroup isomorphic to $\\bZ\\wr\\bZ$. The reason is that, every two--generator subgroup of a RAAG is either free or free abelian~\\cite{Baudisch1981}; see also~\\cite[Corollary 1.3]{KK2015GT}.\nSecond, a RAAG is \\emph{Hopfian}; that is, every endomorphsm of a RAAG is an isomorphism. This follows from a general fact that every finitely generated residually finite group is Hopfian~\\cite{LS2001}.\n\n\n\\bp[Proof of Theorem~\\ref{t:tech-main} from Lemma~\\ref{l:main}]\nAssume $\\form{a,b,t}\\cong\\bZ^2\\ast\\bZ$. \nSince the RAAG $\\bZ^2\\ast\\bZ$ is Hopfian, the natural surjection between groups\n\\[ \\form{A,B,T\\mid [A,B]=1}\\to \\form{a,b,t}\\]\nis actually an isomorphism. It follows that\n\\[\\form{a,b,tat^{-1},tbt^{-1}}\\cong\\form{A,B,TAT^{-1},TBT^{-1}}\\cong\\bZ^2\\ast\\bZ^2.\\]\nThis contradicts Lemma~\\ref{l:main}, since the four diffeomorphism $a,b,tat^{-1},tbt^{-1}$ satisfy the conditions of the lemma.\n\\ep\n\n\\bp[Proof of Lemma~\\ref{l:main}]\nWe put $G=\\form{a,b,c,d}$ and  consider an abstract group\n\\[ G_0 = \\form{a_0,b_0,c_0,d_0\\mid [a_0,b_0]=1=[c_0,d_0]}\\cong \\bZ^2\\ast\\bZ^2.\\]\nThere is a natural surjection $p\\co G_0\\to G$ defined by \\[(a_0,b_0,c_0,d_0)\\mapsto(a,b,c,d).\\] \n\nAssume for a contradiction that $G\\cong G_0$. By the Hopficity of $\\bZ^2\\ast\\bZ^2$,  we see that $p$ is an isomorphism. Since $G$ does not contain $\\bZ\\wr\\bZ$,  Lemma~\\ref{l:global} implies that $G$ has a global fixed point.\nIn other words, we may assume $M=I$.\n\nLet us define \n$\\phi=[c,bdb^{-1}]$ and $\\psi=[\\phi,a]$.\nLemma~\\ref{l:comm-supp} implies that\n\\[\\overline{\\supp\\psi}\\sse\\supp\\phi\\cup\\supp a\\cup\\overline{\\supp\\phi\\cap\\supp a}.\\]\nWe see from Lemma~\\ref{lem:c1} that\n\\[\\overline{\\supp\\phi\\cap\\supp a}\n\\sse\\overline{\\supp\\phi\\setminus\\supp b}\n\\sse\\supp c\\cup \\supp d.\\]\nSo, it follows that\n\\[\n\\overline{\\supp \\psi}\\sse\\supp G.\\]\n\nAs we are assuming $p$ is injective, we have \n\\[\\psi = [\\phi,a]=\\left[ [c, bdb^{-1}],a\\right]\\ne1.\\]\nLemma~\\ref{l:recursive} implies that $G$ contains $\\bZ\\wr\\bZ$, which is a contradiction. This completes the proof.\\ep\n\nLet us conclude this section by describing one generalization of Theorem~\\ref{t:tech-main}.\n\n\\begin{prop}\\label{t:tech-main2}\nLet $M\\in\\{I,S^1\\}$.\nIf $a,b,c,d\\in\\Diff^1_+(M)$ satisfy \n\\[\\supp a\\cap\\supp b=\\varnothing,\\quad\n\\supp c\\cap\\supp d=\\varnothing.\\]\nand \n\\[\n\\left[[c,bdb^{-1}],a\\right]\\ne1,\n\\]\nthen $\\form{a,b,c,d}$ contains the lamplighter group $\\bZ\\wr\\bZ$.\n\\end{prop}\n\nIn particular, the group $\\form{a,b,c,d}$ does not embed into a RAAG.\n\n\n\n\n\n\n\n\\section{Proof of Theorem~\\ref{thm:main}}\\label{s:main-thm}\n\nIn this section, we apply the facts we have gathered to complete the proof of the main result.\n\n\\subsection{Reducing to the connected case}\nWe will reduce the proof of Theorem~\\ref{thm:main} to the case $M\\in\\{I,S^1\\}$, using the following group theoretic observations.\n\n\\begin{lem}\\label{lem:L1}\nSuppose $A,B,C,D$ are groups, and suppose that $A\\times B$ is a normal subgroup of $C*D$. Then at least one of these four groups is trivial.\n\\end{lem}\n\\begin{proof}\nIf $A\\times B$ is a subgroup of $C*D$ then the Kurosh Subgroup Theorem implies that there is a free product decomposition \\[A\\times B\\cong F\\ast\\bigast_{i} H_i,\\] where $F$ is a free group (possibly of infinite rank) and where each $H_i$ is conjugate into $C$ or into $D$. By analyzing centralizers of elements, it is easy to show that a nontrivial free product is never isomorphic to a nontrivial direct product (cf.~\\cite[p.177]{LS2001}). It follows that $A\\times B$ is conjugate into $C$ or $D$, which contradicts the normality of $A\\times B$.\n\\end{proof}\n\nAn alternative proof of Lemma~\\ref{lem:L1} can be given using Bass--Serre theory (see~\\cite{Serre1977}).\n\n\\begin{lem}\\label{lem:L2}\nSuppose $A,B,C,D$ are nontrivial groups, and that $A*B\\le C\\times D$. Then there is an injective homomorphism from $A*B$ into either $C$ or $D$.\n\\end{lem}\n\\begin{proof}\nSuppose the contrary, so that $K_C$ and $K_D$ are the (nontrivial) kernels of the inclusion of $A*B$ into $C\\times D$ composed with the projections onto $C$ and $D$. Then $K_C\\cap K_D=1$ and $K_C$ and $K_D$ normalize each other, so that $K_CK_D\\cong K_C\\times K_D\\le A*B$. This contradicts Lemma~\\ref{lem:L1}.\n\\end{proof}\n\n\n\n\n\n\n\n\\begin{lem}\\label{lem:connected}\nSuppose \\[M=\\coprod_{i=1}^n M_i\\] is a compact one--manifold, and suppose that $A\\ast B$ embeds into $\\Diffb(M)$. Then \n for some finite index subgroups $A_0\\le A$ and $B_0\\le B$,\n  and for some $i$, we have an embedding of $A_0\\ast B_0$ into $\\Diffb(M_i)$.\n\\end{lem}\n\\begin{proof}\nThis follows immediately from Lemma~\\ref{lem:L2}, using the fact that $\\Diffb(M)$ is commensurable with \\[\\prod_{i=1}^n\\Diffb(M_i).\\]\n\nNote that passage to  finite index subgroups is necessary, since $M$ may consist of a union of diffeomorphic manifolds which are permuted by the action of $A\\ast B$.\n\\end{proof}\n\n\\subsection{Taming supports}\n\nLet us denote the center of a group $G$ by $Z_G$.\nIf $M$ is a one--manifold and if $s\\in\\Diffb(M)$, then\nwe denote by $Z(s)$ the centralizer of $s$ in $\\Diffb(M)$.\nThe following lemma is crucial for applying Lemma~\\ref{l:main}:\n\n\n\\begin{lem}\\label{lem:Z2}\nAssume one of the following:\n\\be[(i)]\n\\item\n$M=I$ and $G$ is a nonabelian group such that $Z_G\\ne1$.\n\\item\n$M=S^1$ and $G$ is a non-metabelian group such that $\\bZ\\le Z_G$.\n\\item\n$M=S^1$ and $G$ is a finitely generated group such that $\\bZ\\le Z_G$\nand such that $G$ is not abelian-by-finite cyclic.\n\\ee\nIn each of the cases, if $G\\le\\Diffb(M)$, then there is a subgroup $\\bZ^2\\le G$ generated by diffeomorphisms $a$ and $b$ such that $\\supp a\\cap\\supp b=\\varnothing$.\n\\end{lem}\n\n\n\\bp\n\n\\emph{Case (i).} Let us pick $s\\in Z_G\\setminus1$ and $b\\in[G,G]\\setminus1$. \nSince $G\\le Z(s)$,  Lemma~\\ref{l:interval} implies that \n\\[\\supp s\\cap \\supp b=\\varnothing.\\]\nSince $\\Homeo^+(I)$ is torsion-free, we have $\\form{b,s}\\cong\\bZ^2$ as desired.\n\n\\emph{Case (ii).}\nWe are given with some $s\\in Z_G$ such that $\\form{s}\\cong\\bZ$. As $G$ is nonabelian, Lemma~\\ref{l:circle} (\\ref{p:rot-irr}) implies that $\\rot(s)\\in\\bQ$; in particular, $s^n$ is grounded for some $n\\ge1$.\nFrom part (\\ref{p:hom}) of the same lemma\nand from that $G\\le Z(s)$, we see \nevery element of $[G,G]$ is grounded.\nSince $[G,G]\\le Z(s^n)$, we note from Lemma~\\ref{l:circle} (\\ref{p:zz}) that \n\\[\\supp(s^n)\\cap \\supp G''=\\varnothing.\\]\nFrom the metabelian hypothesis, we can find $b\\in G''\\setminus1$.\nAs $s^n$ and $b$ are grounded, they have infinite orders. It follows that $\\form{s^n,b}\\cong\\bZ^2$.\n\n\n\\emph{Case (iii).}\nLet us proceed similarly to the case (ii). Namely, pick $s\\in Z_G$ such that $\\form{s}\\cong\\bZ$. \nBy Lemma~\\ref{l:circle}, we have a homomorphism\n\\[\n\\rot\\restriction_G\\co G\\to \\bQ.\\]\nWe fix $n\\ge1$ such that $s^n$ is grounded.\nAs $G$ is finitely generated, we see $\\rot(G)$ is finite cyclic.\nThe hypothesis implies that $G_0=\\ker(\\rot\\restriction_G)$ is not abelian.\nSince every element of $G_0$ is grounded, we can apply\n Lemma~\\ref{l:circle} (\\ref{p:zz}) and deduce \n\\[\\supp s^n\\cap \\supp [G_0,G_0]=\\varnothing.\\]\nEach $b\\in [G_0,G_0]\\setminus1$ then yields the desired subgroup\n$\\form{b,s^n}\\cong\\bZ^2$.\n\\ep\n\n\\subsection{The main result}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:main}]\nSuppose $(G\\times\\Z)*\\Z\\le \\Diffb(M)$ for some compact one--manifold $M$. Replacing $G$ by a finite index subgroup if necessary, we may assume that $M$ is connected, by Lemma~\\ref{lem:connected}. \nBy applying the cases (i) and (ii) of Lemma~\\ref{lem:Z2} to the group $G\\times \\bZ$,\nwe can find a subgroup $\\langle a,b\\rangle\\cong\\bZ^2\\le G\\times\\Z$ such that $\\supp a\\cap\\supp b=\\varnothing$.\nIf we write the $\\bZ$--free factor of $(G\\times \\bZ)\\ast\\bZ$ as $\\form{t}$, then\n\\[\\form{a,b,t}\\cong\\form{a,b}\\ast\\form{t}\\cong\\bZ^2\\ast\\bZ.\\]\nThis contradicts Theorem~\\ref{t:tech-main}.\n\\end{proof}\n\nOne can now deduce Corollary~\\ref{cor:fg1} as well as Corollary~\\ref{cor:cnt} below from Lemma~\\ref{lem:Z2}, in the exact same fashion as Theorem~\\ref{thm:main}. \n\\begin{cor}\\label{cor:cnt}\nLet $G$ be a group.\n\\be\n\\item \nIf $G$ is nonabelian and if the center of $G$ is nontrivial,\nthen\n$G*\\Z$ admits no faithful $\\Cb$ action on $I$. \n\\item Suppose $G$ is finitely generated.\nIf $G$ is not abelian-by-finite cyclic\nand if the center of $G$ contains a copy of $\\bZ$,\nthen \n$G*\\Z$ admits no faithful $\\Cb$ action on $S^1$. \n\\end{enumerate}\n\\end{cor}\n\n\n\nHere, a group $G$ is $\\mathcal{X}$--by--$\\mathcal{Y}$ for group theoretic properties $\\mathcal{X}$ and $\\mathcal{Y}$ if there is an exact sequence \\[1\\to K\\to G\\to Q\\to 1\\] such that $K$ has property $\\mathcal{X}$ and $Q$ has property $\\mathcal{Y}$. We allow both $K$ and $Q$ to be trivial.\n\nNote that $G\\times \\bZ$ often occurs as a subgroup of $\\Diffb(M)$, where $G$ is not virtually metabelian. We have the following immediate consequence:\n\n\\begin{cor}\\label{cor:not closed}\nLet $\\mathcal{G}$ denote the class of finitely generated subgroups of $\\Diffb(M)$. The class $\\mathcal{G}$ is not closed under taking finite free products.\n\\end{cor}\n\n\n\n\n\\section{Smooth right-angled Artin group actions on compact one--manifolds}\n\nIn this and the remaining sections, we deduce several corollaries from Theorem \\ref{thm:main}. We first complete the classification of right-angled Artin groups which admit faithful $C^{\\infty}$ actions on a compact one--manifold (Corollary \\ref{cor:classification}).\n\n\\begin{lem}\\label{lem:dichotomy}\nLet $A(\\gam)$ be a right-angled Artin group. Then one of the following mutually exclusive conclusions holds:\n\\begin{enumerate}\n\\item\nWe have $(F_2\\times\\Z)*\\Z\\le A(\\gam)$;\n\\item\nThe graph $\\gam$ lies in $\\mathcal{K}_3$.\n\\end{enumerate}\n\\end{lem}\n\\begin{proof}\nLet us consider a stratification of graph classes:\n\\[ \\KK_2\\sse\\KK_3\\sse\\KK.\\]\n\nSuppose $\\gam\\in\\KK_2$. Then $A(\\gam)$ is the free product of free abelian groups, and hence contains no copy of $(F_2\\times\\Z)*\\Z$.\n\nLet $\\gam\\in\\KK_3\\setminus\\KK_2$. Then $\\gam$ is the join of at least two graphs $\\gam_1,\\gam_2$ in $\\KK_2$. \nWe write\n\\[\nA(\\gam)=A(\\gam_1)\\times A(\\gam_2).\\]\nIf $A(\\gam)$ contains a copy of $(F_2\\times\\bZ)\\ast\\bZ$,\nthen so does $A(\\gam_1)$ or $A(\\gam_2)$\nby Lemma~\\ref{lem:L2}; this would contradict the previous paragraph.\n\nAssume $\\gam\\in\\KK\\setminus\\KK_3$.\nFirst consider the case that $\\gam\\in\\KK_{2i}\\setminus\\KK_{2i-1}$ for some $i\\ge2$. \nWe can write\n\\[\n\\gam=\\coprod_{j=1}^k \\gam_j\\]\nfor some $k\\ge2$ and for some nonempty connected graphs $\\gam_j\\in\\KK_{2i-1}$. \nThese graphs $\\Gamma_j$ cannot all be complete graphs, for otherwise $\\gam\\in\\KK_2$. \nSo at least one graph $\\Gamma_j$ contains $P_3$, the path on three vertices, as a full subgraph. \nThis implies that $A(\\gam)$ contains a copy of $(F_2\\times\\bZ)\\ast\\bZ$.\n\nWe then consider the case that $\\gam\\in\\KK_{2i+1}\\setminus\\KK_{2i}$ for some $i\\ge2$. \nNote $\\gam$ is the join of some graphs $\\gam_1,\\ldots,\\gam_k$ in $\\KK_{2i}$. \nBy the previous graph, each $A(\\gam_j)$ contains $(F_2\\times\\bZ)\\ast\\bZ$.\n\nFinally assume $\\gam\\notin\\KK$, so that $\\gam$ is not a cograph. \nThen we have that $P_4$ is a full subgraph of $\\gam$, so that $A(P_4)\\le A(\\gam)$. The group $A(P_4)$ contains every right-angled Artin group $A(F)$, where $F$ is a finite forest (see~\\cite{KK2013}). Since the defining graph of $(F_2\\times\\Z)*\\Z$ is a copy of a path $P_3$ on three vertices together with an isolated vertex, its defining graph is a finite forest. We see that $(F_2\\times\\Z)*\\Z\\le A(\\gam)$.\n\\end{proof}\n\n\nWe complete the proof of Corollary \\ref{cor:classification} with the following proposition:\n\n\\begin{prop}\\label{prop:cinfty}\nLet $\\gam\\in\\mathcal{K}_3$ and let $M$ be a compact one--manifold. Then there is an embedding of $A(\\gam)$ into $\\Diff_+^{\\infty}(M)$.\n\\end{prop}\n\n\nIn the case when $M=S^1$, we will prove more precise facts in Section \\ref{sec:semiconj}.\n\n\\begin{proof}[Proof of Proposition~\\ref{prop:cinfty}]\nLet $\\Diff^{\\infty}_0(I)$ denote the group of $C^{\\infty}$ diffeomorphisms of the interval which are infinitely tangent to the identity at $\\{0,1\\}$. It suffices to find a copy $A(\\gam)\\le\\Diff^{\\infty}_0(I)$, since $I$ is a submanifold of every compact one--manifold $M$ and every element of $\\Diff^{\\infty}_0(I)$ can, by definition, be extended to all of $M$ by the identity map. \n\nIn the case when $\\gam\\in\\mathcal{K}_2$, we can write \n$A(\\gam)=\\Z^{n_1}\\ast\\cdots\\ast\\Z^{n_k}$.\nAs we have an embedding $A(\\gam)\\hookrightarrow\\Z*\\Z^N$ for $N=\\max_i n_i$, it suffices to prove the proposition for $\\Z*\\Z^N$ in this case. \nTo do this, we first find a copy of $\\Z^N\\le \\Diff^{\\infty}_0(I)$ such that the support of each nontrivial element of $\\Z^N$ is all of $(0,1)$. The existence of such a copy of $\\Z^N$ follows from choosing a $C^{\\infty}$ vector field on $I$ which vanishes only at $\\partial I$ and integrating it to get a flow, which gives an $\\R$--worth of commuting elements of $\\Diff^{\\infty}_0(I)$.\nNow, choosing a generic (in the sense of Baire) element $\\psi$ of $\\Diff^{\\infty}_0(I)$, we have that $\\psi$ and this copy of $\\Z^N$ generate a copy of $\\Z*\\Z^N\\le\\Diff^{\\infty}_0(I)$ (cf.~\\cite{KKMj2016}).\n\n\n\nFinally, if $\\gam\\in\\mathcal{K}_3\\setminus\\mathcal{K}_2$ then $A(\\gam)$ is a finite direct product of $k$ right-angled Artin groups with defining graphs in $\\mathcal{K}_2$. Write again $N$ for the maximal rank of an abelian subgroup of $A(\\gam)$. We choose a finite collection of disjoint intervals $\\{J_1,\\ldots,J_k\\}$ with nonempty interior inside of $I$, and realize a copy of $\\Z*\\Z^N$ on each $J_i$, extending by the identity outside of $J_i$. It is clear that $A(\\gam)$ is thus realized as a subgroup of $\\Diff^{\\infty}_0(I)$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\section{Lower regularity}\n\nIn this section, we prove Proposition \\ref{prop:free prod}. \nRecall a \\emph{left order} on a group $G$ is a total order $\\prec$ on $G$ such that for all triples $a,b,g\\in G$ we have $a\\prec b$ if and only if $ga\\prec gb$. \nA group is \\emph{left orderable} if it admits a left order.\n\n\nEvery subgroup of $\\Homeo^+(\\bR)$ is left orderable. Conversely, if $G$ is countable and left orderable,  then there is a faithful action $G\\to\\Homeo^+(\\bR)$;\nit can be further required from the action that for some fixed point $x_0\\in\\bR$,\nwhenever $g\\prec h$ we have $g(x_0)<h(x_0)$. There exists a standard example of such an action, called a \\emph{dynamical realization} of the given left order~\\cite{Navas2011}.\n\n\nNote that if $\\prec$ is a left order on a group $G$, then $G$ also admits the opposite order $\\prec^{opp}$, where $g\\prec h$ if and only if $h\\prec^{opp} g$.  It follows that if $g$ is a nontrivial element of a countable left orderable group $G$, then there exists a faithful action of $G$ on $\\R$ such that $x_0<g(x_0)$ for some $x_0\\in \\R$. We will call an action coming from the opposite order as an \\emph{opposite action}.\n\nLet us now establish Proposition \\ref{prop:free prod} for the case $M=I$:\n\n\\begin{prop}\\label{prop:homeo closed}\nLet $\\mathcal{G}$ denote the class of countable subgroups of $\\Homeo^+(I)$. Then $\\mathcal{G}$ is closed under countable free products.\n\\end{prop}\n\nProposition \\ref{prop:homeo closed} also follows from the general fact that if $G$ and $H$ are countable left orderable groups then so is $G\\ast H$, as was shown in~\\cite{KM1996,Rivas2012JAlg}. \nWe are including a proof as the idea will be needed for Proposition~\\ref{prop:ab-homeo}.\nUnlike a dynamical realization of $G\\ast H$ coming from its natural left order~\\cite{DNR2014}, the construction below may not have a point $x_0$ with trivial stabilizer.\n\n\\bp\nThe proof is based on the idea of~\\cite{BKK2014}. \nBy induction, it suffices to show that if $G_1,G_2,\\ldots\\in\\mathcal{G}$ then $\\bigast_{i\\ge1}G_i\\in\\mathcal{G}$. Since $K=\\langle \\{G_i\\mid i\\ge1\\}\\rangle\\in\\mathcal{G}$ and since $\\bigast_{i\\ge1}K\\le \\Z*K$ \nby the normal form theorem for free products,\nit suffices to show that if $G\\in\\mathcal{G}$ then $\\Z*G\\in\\mathcal{G}$.\nWe let $H=G\\ast \\bZ$, and choose countably many disjoint subintervals $\\{I_h\\}_{1\\neq h\\in H}$ of $I$, each with nonempty interior. \n\nLet $h\\in H\\setminus1$ be arbitrary.\nWe claim that there exists an action $\\rho_h\\co H\\to\\Homeo^+(I_h)$ such that $h\\not\\in\\ker\\rho_h$.\nFor some $g_i\\in G$ and $r_i\\in\\bZ$, we can write\n\\[ h =g_\\ell t^{r_\\ell} \\cdots g_1t^{r_1} .\\]\nIf $h\\in\\form{t}$ or $h\\in G$, then the claim is obvious.\nSo, possibly after conjugation we may assume that $r_i\\ne0$ and $g_i\\ne1$ for each $i$. \n\nWe choose disjoint closed intervals \\[\\{J_1,\\ldots,J_\\ell\\}\\subset I_h\\] with nonempty interior, $J_i=[p_i,q_i]$ and $q_i<p_j$ for $1\\leq i<j\\leq \\ell$.\nFor each $1\\leq i\\leq \\ell$, we choose an action of $G$ on $J_i$ such that\n $x_i<g_i(x_i)$ for some $x_i\\in J_i$.\n\nWe now define an action of $t$ on $I_h$. We choose $\\ell$ disjoint closed intervals \\[\\{L_1,\\ldots,L_\\ell\\}\\subset I_h\\] of the form $L_i=[c_i,d_i]$. We choose $c_1<x_0<p_1$ and $x_1<d_1<g_1(x_1)$. For $i>1$, we choose points $d_{i-1}<c_i<g_{i-1}(x_{i-1})$ and $x_i<d_i<g_i(x_i)$. We now define $t$ on \\[\\bigcup_{i=1}^\\ell L_i\\] so that $t^{r_1}(x_0)=x_1$ and so that $t^{r_i}(g_{i-1}(x_{i-1}))=x_i$. Since the $\\{L_1,\\ldots,L_\\ell\\}$ are disjoint, such a choice for the definition of $t$ is possible. It is routine to check that $x_0<h(x_0)=g_\\ell(x_\\ell)$. \nSince $h$ is not in the kernel of this action, the claim is proved.\n\nBy taking disjoint subintervals $\\{I_h\\}_{h\\in H\\setminus 1}$ of $\\R$ with each $I_h$ equipped with an action of $H$ by $\\rho_h$, we obtain the desired embedding\n\\[\\rho = \\prod_{h\\in H\\setminus1}\\rho_h \\co H\\to\\prod_{h\\in H\\setminus1}\\Homeo^+(I_h)\\le  \\Homeo^+(\\bR)\\cong\\Homeo^+(I).\\qedhere\\]\n\\ep\n\nWe note from the above proof that the restriction of $G$ on each $J_i=[p_i,q_i]$ corresponds to the original action of $G$, or the opposite action of $G$. \nBy taking $G=\\bZ^2$, we obtain the following.\n\n\\begin{prop}\\label{prop:ab-homeo}\nThere exists an embedding\n\\[ \\rho\\co \\form{a,b,t\\mid [a,b]=1}\\cong\\bZ^2\\ast\\bZ\\to\\Homeo^+(I)\\]\nsuch that $\\supp\\rho(a)$ and $\\supp\\rho(b)$ are disjoint.\n\\end{prop}\nIn particular, the $C^1$ hypothesis in Theorem~\\ref{t:tech-main} cannot be lowered to $C^0$.\n\n\n\n\\section{Complexity of right-angled Artin groups versus diversity of circle actions}\\label{sec:semiconj}\n\nIn this section, we prove Corollary~\\ref{cor:conj}. The proof follows easily from the results in~\\cite{KKMj2016}, together with Corollary~\\ref{cor:classification}.\n\n\nIn a joint work with Mj~\\cite{KKMj2016}, the authors defined a class of finitely generated groups $\\mathcal{F}$ and called each group in the class as \\emph{liftable--flexible}.\nLet us extract the necessary facts which are demonstrated therein.\nRecall from the introduction that a \\emph{projective} action of a group is a representation into $\\PSL(2,\\bR)$.\n\n\\begin{thm}[cf. Theorem 1.1 of~\\cite{KKMj2016}]\\label{thm:KKMj}\nThere exists a class of finitely generated groups $\\FF$ satisfying the following.\n\\be\n\\item For each $G\\in\\FF$, there exist uncountably many distinct semiconjugacy classes of faithful projective actions on $S^1$.\n\\item\nEvery limit group is in $\\FF$; in particular, every free group and every free abelian group is in $\\FF$.\n\\item\nIf $G,H\\in\\FF$, then $G*H\\in\\FF$.\n\\end{enumerate}\n\\end{thm}\n\n\n\n\\begin{lem}\\label{l:ab}\nLet $A$ and $B$ be nontrivial torsion-free finitely generated subgroups such that $B$ is nonabelian. \nWe assume that $A\\times B\\le \\Diffb(S^1)$.\nThen there exist finite index normal subgroups $A_1\\unlhd A$ and $B_1\\unlhd B$ such that \n $A_1\\times B_1$ has a global fixed point on $S^1$.\nFurthermore, the closure of $\\supp A_1$ is a proper subset of $S^1$.\n\\end{lem}\n\\bp\nChoose an arbitrary element $a\\in A\\setminus 1$.\nSince the centralizer $Z(a)$ is nonabelian and $A$ is torsion-free, \nLemma~\\ref{l:circle} implies that $\\rot Z(a)\\sse\\bQ$.\nSince $B$ and $a\\times B$ are subsets of $Z(a)$, we see that $\\rot(B), \\rot(a\\times B)\\sse\\bQ$.\nThis shows that $\\rot(A\\times B)\\sse\\bQ$.\n \nFor each $a\\in A$ and $b\\in B$, note that \n \\[a,b,ab\\in Z(a).\\]\nBy Lemma~\\ref{l:circle} (\\ref{p:hom}), it follows that $\\rot(a)+\\rot(b)=\\rot(ab)$.\nIn particular, we have a homomorphism \n\\[\\rot\\co A\\times B\\to \\bQ.\\]\nSince $A\\times B$ is finitely generated, the image of the above map is finite.\nSo we can find finite index normal subgroups $A_1\\unlhd A$ and $B_1\\unlhd B$ such that every element of $A_1\\times B_1$ is grounded. \n\nLet $a\\in A_1\\setminus1$.\nSince $B_1$ centralizes $a$, it acts on each component of $\\supp a$ as an abelian group; see Lemma~\\ref{l:disj-abel}. It follows that $B_1$ preserves the set $\\supp A_1$ and\n\\[\\supp A_1\\cap\\supp [B_1,B_1]=\\varnothing.\\]\nSince $B_1$ is nonabelian, we see $\\overline{\\supp A_1}\\ne S^1$. \nIt follows that $\\partial\\supp A_1$ is a global fixed point of $A_1\\times B_1$.\n\\ep\n\n\\begin{proof}[Proof of Corollary~\\ref{cor:conj}]\n\n(1) \nIf $\\gam\\in\\mathcal{K}_2$, then $A(\\gam)$ is a finite free product of free abelian groups. Combining parts (2) and (3) of Theorem~\\ref{thm:KKMj}, we see $A(\\gam)$ is in the class $\\FF$. The  first part of the same theorem implies the desired conclusion.\n\n(2) Suppose that $\\gam\\in\\mathcal{K}_3\\setminus\\mathcal{K}_2$, so that $A(\\gam)$ decomposes as a nontrivial direct product $A(\\gam_1)\\times A(\\gam_2)$, where at least one of $A(\\gam_1)$ and $A(\\gam_2)$ is nonabelian. Say $A(\\gam_2)$ is nonabelian. \nBy Lemma~\\ref{l:ab}, there exist finite index normal subgroups $G\\unlhd A(\\gam_1)$\nand $H\\unlhd A(\\gam_2)$ \nsuch that $G\\times H$ has a global fixed point; in particular, $A(\\gam)$ has a finite orbit. \nWe put\n\\[X = \\overline{\\supp G}, \\quad Y = S^1\\setminus X.\\]\nSince $G$ is normal in $A(\\gam)$ we see that \n$X$ and $Y$ are $A(\\gam)$--invariant sets of $S^1$, which are proper by the same lemma.\nThis implies that $X$ and $Y$ have nonempty interiors, and hence $A(\\gam)$ does not have a dense orbit. \n\n\n\n(3) This follows immediately from Corollary~\\ref{cor:classification}.\n\\end{proof}\n\nWe briefly remark that if a group acts on $S^1$ with a global fixed point then it is semi--conjugate to a trivial action, and if it acts with a periodic point then the action is semi--conjugate to a rational rotation group. Corollary~\\ref{cor:conj} implies that if $\\gam\\in\\mathcal{K}_3\\setminus\\mathcal{K}_2$ then $A(\\gam)$ admits only countably many semi--conjugacy classes of faithful actions, and it is not difficult to realize one semi--conjugacy class for each rational rotation.\n\nObserve that since every right-angled Artin group surjects to $\\Z$, every right-angled Artin group admits uncountably many distinct semi--conjugacy classes of non--faithful actions on $S^1$, so only faithful actions are interesting for our purposes.\n\n\n\n\n\n\n\n\n\n\n\\section{Thompson's groups}\nLet $F$ and $T$ be the Thompson's groups acting on the closed interval and on the circle, respectively~\\cite{CFP1996}. Recall that these are the groups of piecewise linear homeomorphisms of the interval and circle respectively, with dyadic breakpoints and all slopes given by powers of two. It is known that the standard action of $T$ is conjugate to a $C^\\infty$ action~\\cite{GS1987}.\nThe restriction of such a smooth action yields a smooth action of $F$ on a closed interval.\nOn the other hand, the groups $F*\\Z$ and $T\\ast\\bZ$ do not admit any smooth actions on the circle, or indeed on any compact one--manifold (cf. Corollary~\\ref{cor:Thompson}):\n\n\\begin{cor}\nIf $M$ is a compact one-manifold,\nthen $F\\ast\\bZ$ and $T\\ast\\bZ$ are not subgroups of $\\Diff^{1+\\mathrm{bv}}(M)$.\n\\end{cor}\n\n\\bp\nSince $F\\le T$, it suffices to prove the corollary for $F*\\Z$ only. In order to apply Theorem~\\ref{thm:main}, it suffices to show that $F\\times\\Z\\le F$ and that $F$ is not virtually metabelian, whence the conclusion will be immediate. These claims follow immediately from the well--known facts that $F$ is not virtually solvable and that $F\\times F\\le F$, and we make these details explicit below for the reader's convenience.\n\nSince $F$ is a group of homeomorphisms of the interval, it is immediate that it is torsion--free. Moreover, conjugating $F$ by the homeomorphism of $\\R$ given by $x\\mapsto x\/2$ scales $F$ to be the group of piecewise linear homeomorphisms with dyadic breakpoints and slopes given by powers of two, only scaled to act on the interval $[0,1\/2]$. It follows that we may realize $F\\times F\\le F$, since we can realize one copy of $F$ on $[0,1\/2]$ and a second one on the interval $[1\/2,1]$, with the points $\\{0,1\/2,1\\}$ globally invariant. It follows that $\\Z\\times F\\le F$, since $F$ is torsion--free.\n\nTo see that $F$ is not virtually metabelian, we use the standard fact that $[F,F]$ is an infinite simple group and that $Z(F)=\\{1\\}$ (cf.~\\cite{CFP1996}). It follows that if $H\\le F$ is a finite index subgroup then $[F,F]\\le H$. Since $H$ contains an infinite simple group, it cannot be solvable, much less metabelian.\n\\ep\n\n\n\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\nThe authors thank the anonymous referees for many careful comments which improved the exposition of the paper. The first author is supported by Samsung Science and Technology Foundation (SSTF-BA1301-06).\nThe second author is partially supported by Simons Foundation Collaboration Grant number 429836.\n\n\n\\def\\cprime{$'$} \\def\\soft#1{\\leavevmode\\setbox0=\\hbox{h}\\dimen7=\\ht0\\advance\n  \\dimen7 by-1ex\\relax\\if t#1\\relax\\rlap{\\raise.6\\dimen7\n  \\hbox{\\kern.3ex\\char'47}}#1\\relax\\else\\if T#1\\relax\n  \\rlap{\\raise.5\\dimen7\\hbox{\\kern1.3ex\\char'47}}#1\\relax \\else\\if\n  d#1\\relax\\rlap{\\raise.5\\dimen7\\hbox{\\kern.9ex \\char'47}}#1\\relax\\else\\if\n  D#1\\relax\\rlap{\\raise.5\\dimen7 \\hbox{\\kern1.4ex\\char'47}}#1\\relax\\else\\if\n  l#1\\relax \\rlap{\\raise.5\\dimen7\\hbox{\\kern.4ex\\char'47}}#1\\relax \\else\\if\n  L#1\\relax\\rlap{\\raise.5\\dimen7\\hbox{\\kern.7ex\n  \\char'47}}#1\\relax\\else\\message{accent \\string\\soft \\space #1 not\n  defined!}#1\\relax\\fi\\fi\\fi\\fi\\fi\\fi}\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]{%\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \nBerkovich analytic spaces and rigid analytic spaces \\cite{Ber1990}, \\cite{Ber2009} have the advantage that both schemes of finite type over a field and formal completions of such schemes along their closed subschemes can be thought of as living in the same category of analytic spaces. Punctured tubular neighborhoods of algebraic subvarieties inside ambient varieties over any field can be defined \\cite{BeTe}, \\cite{Th} as Berkovich analytic spaces.  In this article, we consider non-Archimedean analytic spaces from the perspective of algebraic geometry relative to the closed symmetric monoidal categories of Banach spaces. This language is very universal and provides a place to compare different geometries (rigid analytic spaces, Berkovich spaces and others). To\\\"{e}n and Vaqui\\'{e} introduced in \\cite{TV} algebraic geometry relative to any closed symmetric monoidal category. This idea had also been pursued by Hakim \\cite{Ha} and Deligne \\cite{Del}. We examine one of To\\\"{e}n and Vezzosi's \\cite{TVe5} topologies (which we call the homotopy Zariski topology) restricted to the opposite category of affinoid algebras over a non-Archimedean field and show that Berkovich and rigid analytic geometry embed fully and faithfully into the resulting category of schemes. \nWe use the framework of To\\\"{e}n, Vaqui\\'{e} and Vezzosi to suggest a new approach to analytic geometry which will extend in a natural way to the setting of higher and derived analytic stacks over Banach rings. This of course produces the difficult task of comparing the abstract definitions to the existing ones.  In order to make this article accessible to a broad range of mathematicians, we have included as many details as was possible. In forthcoming work \\cite{BaBe}, \\cite{BaBeKr}, we incorporate complex analytic geometry and give a more general approach that covers dagger analytic geometry and Stein geometry over a general valuation field (Archimedean or non-Archimedean). Some of the properties of the localization maps which we look at were proven from a complex analytic or differential geometric point of view in \\cite{Block, Pir, Tay}. Some similar work to ours was done \\cite{Mac} by A. Macpherson who uses an abstract, categorical notion of localizations instead of the condition on derived categories which we use. See also \\cite{Yu} and the references therein for the relationship with mirror symmetry. \n\nUsing the homological algebra from the work of Schneiders \\cite{SchneidersQA}, we give a new interpretation of the homotopy Zariski topology from \\cite{TVe5} which is suited to deal with quasi-abelian categories. Two of the theorems which we prove (Theorems \\ref{thm:TensLocs} and \\ref{thm:localForm}) show that that the finitely presented morphisms of affinoid algebras $\\mathcal{A}\\to \\mathcal{B}$ which are \\emph{homotopy epimorphisms} (meaning that $\\mathcal{B}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B} \\cong \\mathcal{B}$) correspond geometrically to the affinoid subdomains (affinoids which are the unions of rational domains). In terms of modules, this says that a morphism of affinoids $\\mathcal{M}(\\mathcal{B}) \\to \\mathcal{M}(\\mathcal{A})$ is a domain embedding if and only if the corresponding morphism $D^{-}(\\mathcal{B})\\to D^{-}(\\mathcal{A})$ is fully faithful. Our categorical characterization of affinoid domains answers an open question poised by Soibelman in section 1.4 of \\cite{So}. He identifies this question as being a key step in the development of non-commutative analytic geometry. The other main feature of the G-topology is that the covers in this topology have finite subcovers $\\pi:\\coprod_{i} \\mathcal{M}(\\mathcal{A}_{V_i}) \\to \\mathcal{M}(\\mathcal{A})$ of an affinoid by affinoid subdomains which are surjective. In order to understand this surjectivity via modules, we introduce the notion of a RR-quasicoherent module for a commutative monoid $A$ relative to a symmetric monoidal quasi-abelian category in Definition \\ref{defn:RRqc}. We call this category of modules ${\\text{\\bfseries\\sf{Mod}}}^{RR}(A)$.   We prove in Lemmas \\ref{lem:ConsImpliesSur} and \\ref{lem:CoversRconservative} that $\\pi:\\coprod_{i} \\mathcal{M}(\\mathcal{A}_{V_i}) \\to \\mathcal{M}(\\mathcal{A})$ is surjective if and only if a morphism $f:M \\to N$ of RR-quasicoherent modules is an isomorphism if and only $\\pi^{*}f: \\pi^{*}M \\to \\pi^{*}N$ is an isomorphism. These modules satisfy a version of Tate acyclicity (see Remark \\ref{rem:TateAcylicity}) for the version we use of the Tate complex which uses only completed tensor products. More work on the category of quasi-coherent sheaves in analytic geometry will appear in \\cite{BaBeKr, BeKr2}.\n\nThe condition we use on morphisms of algebras: homotopy epimorphism, is the key idea behind the holomorphic functional calculus studied by \\cite{Tay} by Taylor in complex analytic geometry, who called these morphisms absolute localizations. Pirkovskii in \\cite{Pir} shows that open embeddings of complex Stein manifolds satisfy the same abstract condition. \n\n\n In \\cite{BeKr2} we construct monoidal model structures on categories of simplicial objects and on categories of complexes (negatively graded or unbounded) in a quasi-abelian category ${\\tC}$ satisfying certain extra conditions. We prove in \\cite{BeKr2} that these structures form HAG contexts in the sense of \\cite{TVe3}.  A HAG context is essentially a monoidal model category $M$ satisfying a long list of axioms that make it easy to work with. The model structure is induced from the model structure on simplicial sets. For instance, a morphism $X_{\\bullet} \\to Y_{\\bullet}$ in the category $M={\\text{\\bfseries\\sf{sC}}}$ of simplicial objects in ${\\tC}$  is a fibration (resp. weak equivalence) if for all projectives $P$ in ${\\tC}$ (defined in \\ref{defn:Projective}) that $\\Hom(P, X_{\\bullet}) \\to \\Hom(P, Y_{\\bullet})$ is a fibration (resp. weak equivalence) of simplicial sets. In particular, our main application is when the quasi-abelian category ${\\tC}={\\text{\\bfseries\\sf{Ind}}}({\\text{\\bfseries\\sf{Ban}}}_{R})$ is the category of Ind-Banach spaces over a Banach ring. In \\cite{TVe3} derived and homotopy algebraic geometry is developed relative to a HAG context.  Derived algebraic geometry (the case where ${\\tC}$ is the abelian category of abelian groups) is only one special case of their work. Our goal is to show that derived analytic geometry is another. A key feature in their work is definitions of (Grothendieck) topologies on the categories of affine schemes relative to $M$. Affine schemes are defined to be opposite category to ${\\text{\\bfseries\\sf{Comm}}}(M)$, the category of commutative monoids relative to $M$. The main way to describe these topologies is to explain which morphisms should appear in the cover, and describe when such a collection of morphisms is a cover. Both of these can be explained in terms of geometrically induced functors on the homotopy categories of modules. One topology that can be defined from their point of view says \n\\begin{defn}\\label{defn:TVtopologyIntro}\n A cover of $\\spec(A) \\in {\\text{\\bfseries\\sf{Comm}}}(M)^{op}$ is a collection of morphisms $\\spec(B_i) \\to \\spec(A)$ in ${\\text{\\bfseries\\sf{Comm}}}(M)^{op}$ for $i \\in I$ such that for some finite set $J \\subset I$ we have \n\\begin{enumerate}\n\\item the push forward functor from the derived category of modules on $B_{i}$ to that on $A$ is fully faithful for all $i \\in J$\n\\item a morphism in the derived category of modules on $A$ is an isomorphism if and only if it becomes an isomorphism when pulled back in the derived sense to a morphism in the derived category of modules on $B_i$ for all $i \\in J$.\n\\end{enumerate} \nFor more details see Definition 1.2.6.1 (3) of \\cite{TVe3} for the first item and Definition 1.2.5.1 of \\cite{TVe3} for the second.\n\\end{defn}\nIn our type of derived analytic geometry, the category of affine schemes over a valuation field is opposite to the category of commutative monoids relative to $M={\\text{\\bfseries\\sf{sInd}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$. Theorem \\ref{thm:BigRestriction} shows that on the subcategory opposite to affinoid algebras, the topology in Definition \\ref{defn:TVtopologyIntro} restricts to the weak G-topology on affinoids. This theorem relies on \\cite{BeKr2}.\n\nWhile condition (1) of Definition \\ref{defn:TVtopologyIntro} restricts by our results to something completely classical and well understood, Condition (2) of Definition \\ref{defn:TVtopologyIntro} seems difficult to check on modules. Therefore, one might try to use a similar condition on the (underived) quasi-abelian category of modules using the underived pullback. However, this does not correspond to the surjectivity condition on covers, forcing us to reconsider which modules we care about and introduce the RR-quasicoherent modules.  For this reason, we defined the formal homotopy Zariski topology on a subcategory ${\\text{\\bfseries\\sf{A}}}$ of the category of affine schemes (${\\text{\\bfseries\\sf{Comm}}}({\\tC})^{op}$) of a closed symmetric monoidal quasi-abelian category ${\\tC}$ where ${\\text{\\bfseries\\sf{A}}}$ and ${\\tC}$ satisfy a few simple conditions. The covers are a collection of morphisms $A\\to B_{i}$ of in ${\\text{\\bfseries\\sf{A}}}$ for $i \\in I$ such that for some finite set $J \\subset I$\n\\begin{enumerate}\n\\item $A \\to B_{i}$  are homotopy epimorphisms in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ for all $i \\in J$\n\\item a morphism $f:M \\to N$ in ${\\text{\\bfseries\\sf{Mod}}}^{RR}(A)$ is an isomorphism if and only if the induced morphisms $M\\overline{\\otimes}_{A}B_i \\to N\\overline{\\otimes}_{A}B_i$ are for all $i \\in J$.\n\\end{enumerate}\nIn the special case that ${\\tC}={\\text{\\bfseries\\sf{Ban}}}_{k}$ and ${\\text{\\bfseries\\sf{A}}}$ is the category of affinoids our Theorem \\ref{thm:covers} says that this topology restricts to the weak G-topology on affinoids. Our work, put together with the work of To\\\"{e}n and Vezzosi establishes foundations for derived (and homotopy) analytic geometry and the study of higher and derived stacks over general Banach rings. There is another approach due to F. Paugam. In the complex analytic case, J. Lurie and M. Porta have a different approach to derived analytic geometry which should be compared to ours. \n\n\n\\section{Algebras and modules in closed symmetric monoidal categories} We assume that the reader is comfortable with the idea of a symmetric monoidal category. In particular the bifunctor $\\overline{\\otimes}$ admits an adjoint  $\\underline{\\Hom} :{\\tC}^{op} \\times {\\tC} \\to {\\tC}$ in the sense that there are natural isomorphisms\n\\[\\Hom(U \\overline{\\otimes} V,W) \\cong \\Hom(U, \\underline{\\Hom}(V,W))\\] for any $U,V,W \\in {\\tC}.$\nConsider a category ${\\tC}$ which has all finite limits and colimits equipped with extra data $(\\underline{\\Hom}, \\overline{\\otimes}, \\text{id})$ making it into a closed, symmetric monodial category. \nIn such a category one always has the following natural isomorphisms for any $U,V,W \\in {\\tC}$\n\n\\begin{enumerate}\n\\item \\[\\Hom(U,V) \\cong \\Hom(\\text{id}, \\underline{\\Hom}(U,V))\\]\n\\item \\[\\underline{\\Hom}(U \\overline{\\otimes} V,W) \\cong \\underline{\\Hom}(U, \\underline{\\Hom}(V,W))\\]\nwhich lifts by taking the $\\Hom$ from $\\text{id}$ the corresponding isomorphisms for $\\Hom$ instead of $\\underline{\\Hom}$\n\\item \\[ U \\cong \\underline{\\Hom}(\\text{id},U).\\]\n\\end{enumerate}\n\n\nAlso, there are natural morphisms for any $T,U,V,W \\in {\\tC}$\n\n\\begin{enumerate}\n\\item \\[\\underline{\\Hom}(U,U) \\overline{\\otimes} U \\to U\\]\n\\item \\[\\underline{\\Hom}(V,W) \\overline{\\otimes} \\underline{\\Hom}(U,V) \\to \\underline{\\Hom}(U,W) \\] satisfying associativity \n\\item \\[\\underline{\\Hom}(T,U) \\overline{\\otimes} \\underline{\\Hom}(V,W) \\to \\underline{\\Hom}(T \\overline{\\otimes} V, U \\overline{\\otimes} W)  \\]\n\\item \\[\\text{id} \\to \\underline{\\Hom}(U,U)\\]\n\\end{enumerate}\n\nwhich are compatible with the corresponding morphisms for $\\Hom$ instead of $\\underline{\\Hom}$. \n\\begin{rem}\nThe morphisms in the second item are adjoint to morphisms \n\\[\\underline{\\Hom}(V,W) \\to \\underline{\\Hom}( \\underline{\\Hom}(U,V) ,\\underline{\\Hom}(U,W))\n\\]\nand\n\\[\\underline{\\Hom}(U,V) \\to \\underline{\\Hom}( \\underline{\\Hom}(V,W) ,\\underline{\\Hom}(U,W))\n\\]\nand hence maps\n\\[\\Hom(V,W) \\to \\Hom( \\underline{\\Hom}(U,V) ,\\underline{\\Hom}(U,W))\n\\]\nand\n\\[\\Hom(U,V) \\to \\Hom( \\underline{\\Hom}(V,W) ,\\underline{\\Hom}(U,W)).\n\\]\n\\end{rem}\nSuch categories admit clear definitions of categories of commutative unital algebras objects in them. We denote these categories by ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$. The opposite category to ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ we will be denoted ${\\text{\\bfseries\\sf{Aff}}}({\\tC})= {\\text{\\bfseries\\sf{Comm}}}({\\tC})^{op}$ and if $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$ we use $\\spec(A)$ to indicate this object in the opposite category.\n\n\nFor any fixed commutative unital algebra object $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$ there is a category of unital left modules over such an algebra which we denote ${\\text{\\bfseries\\sf{Mod}}}(A)$. See Chapter 1.3 of \\cite{M} for details.  \nBecause we will be considering the unital notions only, we drop the word unital from our descriptions. This category is also a  closed symmetric monoidal category and has all finite limits and colimits. We write $\\overline{\\otimes}_{A}$ for the monoidal structure and $\\Hom_{A}$ for the morphisms in this category.  Given a morphism $p:\\spec(B) \\to \\spec(A)$, we sometimes write $p^{*}$ for the functor \n\\[{\\text{\\bfseries\\sf{Mod}}}(A) \\to {\\text{\\bfseries\\sf{Mod}}}(B)\n\\]\ngiven by\n\\[\nM \\mapsto B \\overline{\\otimes}_{A} M. \n\\]\n\\begin{defn}\\label{defn:TensA}Let us consider $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$ and $E,F \\in {\\text{\\bfseries\\sf{Mod}}}(A)$. Let $a_{E}:A \\overline{\\otimes} E \\to E$ and $a_{F}:A \\overline{\\otimes} F \\to F$ denote the action morphisms. \nThe set of morphisms $\\Hom_{A}(E,F)$ is defined as the limit of the diagram\n\n\\[\n\\xymatrix{\\Hom(E,F) \\ar@\/^2pc\/[rr]^{f \\mapsto f \\circ a_{E}} \\ar[dr]_{h \\mapsto \\text{id}_{A} \\otimes h} &   & \\Hom(A \\overline{\\otimes} E,  F) \\\\\n& \\Hom(A \\overline{\\otimes} E, A \\overline{\\otimes} F)\\ar[ur]_{g \\mapsto a_{F} \\circ g} & \n}.\n\\] \nIn order to describe the tensor product, let us use $\\sigma:A \\overline{\\otimes} F \\to F \\overline{\\otimes} A$ to denote the symmetric structure. Let $E \\overline{\\otimes}_{A} F\\in {\\text{\\bfseries\\sf{Mod}}}(A)$ be the element of ${\\tC}$ given as the colimit of the diagram \n\\begin{equation}\\label{equation:TensDef}\\xymatrix{E \\overline{\\otimes} A \\overline{\\otimes} F \\ar@\/^\/[rr]^{\\text{id}_{E} \\otimes a_{F}} \\ar@\/_\/[rr]_{(a_{E} \\otimes \\text{id}_{F})\\circ (\\sigma \\otimes \\text{id}_{F})} & & E \\overline{\\otimes} F}\n\\end{equation}\nendowed with the obvious action of $A$.\n\n\\end{defn}\nNotice that for any $E\\in {\\text{\\bfseries\\sf{Mod}}}(A)$ and $F \\in {\\tC}$ can define \n\\[L_{E,F}:A\\overline{\\otimes} \\underline{\\Hom}(E,F) \\to  \\underline{\\Hom}(E,F)\n\\]\nas the composition \n\\[A\\overline{\\otimes} \\underline{\\Hom}(E,F) \\to  \\underline{\\Hom}(E,E) \\overline{\\otimes} \\underline{\\Hom}(E,F)\\to  \\underline{\\Hom}(E,F)\n\\]\nwhere the morphism $A \\to \\underline{\\Hom}(E,E)$ is adjoint to the action morphisms $A \\overline{\\otimes} E \\to E.$ Similarly for any $E\\in {\\tC}$ and $F \\in {\\text{\\bfseries\\sf{Mod}}}(A)$, one can define a morphism \n\\[R_{E,F}:A \\overline{\\otimes} \\underline{\\Hom}(E,F)  \\to  \\underline{\\Hom}(E,F)\n\\]\nas the composition \n\\[ A \\overline{\\otimes} \\underline{\\Hom}(E,F)  \\to  \\underline{\\Hom}(F,F) \\overline{\\otimes} \\underline{\\Hom}(E,F) \\to   \\underline{\\Hom}(E,F) \\overline{\\otimes} \\underline{\\Hom}(F,F)  \\to \\underline{\\Hom}(E,F).\n\\]\nBoth $L_{E,F}$ and $R_{E,F}$ endow  $\\underline{\\Hom}(E,F)$ with the structure of an element of ${\\text{\\bfseries\\sf{Mod}}}(A)$ satisfying various natural properties and we use these structures without further comment.\n\\begin{lem}\\label{lem:ALinHom}Suppose now that ${\\tC}$ is a closed, symmetric monoidal additive category with all finite limits and colimits and $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$. Then ${\\text{\\bfseries\\sf{Mod}}}(A)$ is a closed symmetric monoidal category with all finite limits and colimits as well. These limits and colimits can be computed in ${\\tC}$.\n\\end{lem}\n{\\bf Proof.}\nBy tensoring the morphisms $\\text{id} \\to \\underline{\\Hom}(A,A)$ with the identity of $\\underline{\\Hom}(E,F)$ one can form \n\\[\\underline{\\Hom}(E,F) \\to \\underline{\\Hom}(A,A) \\overline{\\otimes} \\underline{\\Hom}(E,F) \\to \\underline{\\Hom}(A\\overline{\\otimes} E,A\\overline{\\otimes} F). \n\\] Now by using the functor $\\underline{\\Hom}(A\\overline{\\otimes} E, \\;)$ and the contravariant functor $\\underline{\\Hom}(\\;,F)$ applied to $A\\overline{\\otimes} F \\to F$ and $A\\overline{\\otimes} E \\to E$ respectively one can define $\\underline{\\Hom}_{A}(E,F)$ with the same type of limit as in \\ref{defn:TensA} replacing $\\Hom$ with $\\underline{\\Hom}$. Both $L_{E,F}$ and $R_{E,F}$ induce well defined (and equal) morphisms on  $\\underline{\\Hom}_{A}(E,F)$ and give  $\\underline{\\Hom}_{A}(E,F)$ the structure of an element of ${\\text{\\bfseries\\sf{Mod}}}(A)$.\n\\ \\hfill $\\Box$\n\nWe have natural isomorphisms \n\\begin{equation}\\label{equation:ClosedPlus}\\underline{\\Hom}(M \\overline{\\otimes}_{A} N, L) \\cong \\underline{\\Hom}(M, \\underline{\\Hom}_{A}(N,L))\n\\end{equation}\nwhich satisfy the relevant axioms of a closed, symmetric monoidal category.\nIt is easy to see that (as in) \\cite{MeyerDerived} for any $E\\in {\\text{\\bfseries\\sf{Mod}}}(A)$\n\\begin{lem}\\label{lem:MeyerProperties}\\[\\underline{\\Hom}_{A}(A,E) \\to E \\]\n\\[f \\mapsto f(1)\n\\]\nis an isomorphism and so for any modules $M,N \\in {\\text{\\bfseries\\sf{Mod}}}(A)$ and any $V \\in {\\tC}$ we have \n\\begin{enumerate}\n\\item $A \\overline{\\otimes}_{A} M \\cong M$\n\\item $\\underline{\\Hom}_{A}(A \\overline{\\otimes} V,E) \\cong \\underline{\\Hom}(V,E)$\n\\item $\\underline{\\Hom}_{A}(E, \\underline{\\Hom}(A,V)) \\cong \\underline{\\Hom}(E,V)$\n\\item $\\Hom_{A}(A, \\underline{\\Hom}_{A}(M,N)) \\cong \\Hom_{A}(M,N)$\n\\end{enumerate} \n\\end{lem}\n\nWe will discuss in this article various closed symmetric monoidal categories whose objects are vector spaces over Archimedean or non-Archimedean fields equipped with extra structures. The unit object in these categories is the field itself and it is easy to check the associativity, commutativity, and unit constraints. It is also clear that all of our closed symmetric monoidal categories are $k$-linear additive categories, as are the categories ${\\text{\\bfseries\\sf{Mod}}}(A)$ for $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$. Because ${\\tC}$ has all finite limits and colimits, ${\\text{\\bfseries\\sf{Mod}}}(A)$ does as well. \n\n\\begin{defn}A functor $\\straightF:{\\tT}\\to {\\tS}$ is \nsaid to be conservative under that \nthe condition that a morphism $f$ in \n${\\tT}$ is an isomorphism if and only \nif $\\straightF(f)$ is an isomorphism.\n\\end{defn}\nNotice that for every $p:\\spec (B) \\to \\spec (A)$ in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ the functor $p^{*}:{\\text{\\bfseries\\sf{Mod}}}(A) \\to {\\text{\\bfseries\\sf{Mod}}}(B)$ has a conservative right adjoint given by considering a $B$ module as an $A$ module \n\\[p_{*}:{\\text{\\bfseries\\sf{Mod}}}(B) \\to {\\text{\\bfseries\\sf{Mod}}}(A).\\]\n\n\\begin{defn}\\label{Symmetric}\n\nIf ${\\tC}$ has countable coproducts, the forgetful functor $\\straightF:{\\text{\\bfseries\\sf{Comm}}}({\\tC})\\to {\\tC}$ has a left adjoint $\\straightS:{\\tC}\\to {\\text{\\bfseries\\sf{Comm}}}({\\tC})$, given on objects \nby \n\\begin{equation*}\n\\straightS(V)=\\coprod_{n\\geq 0} \\straightS^n(V)\n\\end{equation*}\n\nwhere $\\straightS^n(V)$ are the co-invariants for the symmetric group action on $V^{\\overline{\\otimes} n}$. If $A$ is a commutative algebra in ${\\tC}$ we get in a similar way a functor \n$\\straightS_A:{\\text{\\bfseries\\sf{Mod}}}(A) \\to {\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Mod}}}(A))$ which is left adjoint to the forgetful functor. \n\\end{defn}\n\n\\begin{defn}\\label{defn:morphismSymAlg}\nLet ${\\tC}$ be a closed symmetric monoidal category with countable coproducts. Let $E$ and $E'$ be two objects, and assume given a map of algebras $\\alpha:\\straightS(E')\\to \\straightS(E)$. Let us denote by $\\straightS(E,E',\\alpha)$ the quotient of $\\straightS(E)$ by the ideal generated by the image of $\\alpha$ restricted to the augmentation ideal. In the case that $A\\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$ and $E$ and $E'$ are in ${\\text{\\bfseries\\sf{Mod}}}(A)$ we use the notation $\\straightS_{A}(E,E',\\alpha)$ to denote the same construction taken in the closed symmetric monoidal category ${\\text{\\bfseries\\sf{Mod}}}(A)$ in place of ${\\tC}$.\nAn equivalent way of describing it is as the quotient of $\\straightS(E)$ by the ideal generated by the image of the object $E'$ under the map $\\alpha$.\n\\end{defn}\n \n \\hfill $\\Box$\n\\section{Algebraic geometry relative to closed symmetric monoidal categories}\\label{TV}\n\nIn this section, we review several notions from the article \\cite{TV} by To\\\"{e}n and Vaqui\\'{e}. We review their definitions of a flat morphism, a Zariski open immersion, and the Zariski and fpqc topologies on the opposite of the category of algebra objects in a closed, symmetric monoidal category admitting all finite limits and colimits. We also include definitions of a formal Zariski open immersion and the formal Zariski topology. We also review the definitions of schemes and (higher) stacks in this setting.\n\\begin{defn}\nThe category of presheaves of sets on ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is the category whose objects are contravariant functors ${\\text{\\bfseries\\sf{Aff}}}({\\tC}) \\to {\\text{\\bfseries\\sf{Set}}}$ and whose morphisms are natural transformations. It will be denoted ${\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Aff}}}({\\tC}))$. Given a Grothendieck topology $T$ on the category ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ we will often consider the full subcategories of sheaves of sets ${\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Aff}}}({\\tC})^{T}) \\subset {\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Aff}}}({\\tC}))$, where ${\\text{\\bfseries\\sf{Aff}}}({\\tC})^{T}$ is the site consisting of the underlying category ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ along with the Grothendieck topology $T$.\n\\end{defn}\n\\begin{defn}\nA family of functors $\\{\\straightF_{i}:{\\tC} \\to \\text{\\bfseries\\sf{D}}_{i}\\}_{i \\in I}$ \nis said to be conservative if a morphism $f$ in ${\\tC}$ \nis an isomorphism if and only if the \nmorphisms $\\straightF_{i}(f)$ in $\\text{\\bfseries\\sf{D}}_i$ are \nisomorphisms for all $i \\in I$.\n\\end{defn}\nSince we do not assume the existence of arbitrary limits and colimits, we need to define finite presentation in a non-standard way.\n\\begin{defn}\\label{defn:AbstractFinitePres}Let ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. An object $W\\in {\\tC}$ is called of finite presentation if for every filtered diagram of objects $V_{i} \\in {\\tC}$ such that $\\colim_{i}V_{i}$ exists, the natural morphism\n\\begin{equation}\\label{eqn:AbstractFinitePres}\\colim_{i}\\Hom_{{\\tC}}(W,V_{i}) \\to \\Hom_{{\\tC}}(W,\\colim_{i}V_{i})\n\\end{equation}\nis an isomorphism.\n\\end{defn}\n\\begin{rem}Notice that any finite colimit of finitely \npresented objects is finitely presented.\n\\end{rem}\n\\begin{defn}\\label{defn:FinitePres}\nLet ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. A morphism $A\\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is of finite presentation if for every filtered diagram of objects $A'_{i}\\in A\/{\\text{\\bfseries\\sf{Comm}}}({\\tC})$ such that $\\colim_{i}A_{i}'$ exists in $A\/{\\text{\\bfseries\\sf{Comm}}}({\\tC})$, the natural morphism\n\\begin{equation}\\label{eqn:FinitePres}\\colim_{i}\\Hom_{A\/{\\text{\\bfseries\\sf{Comm}}}({\\tC})}(B,A_{i}') \\to \\Hom_{A\/{\\text{\\bfseries\\sf{Comm}}}({\\tC})}(B,\\colim_{i}A_{i}')\n\\end{equation}\nis an isomorphism.\n\\end{defn}\n\\begin{rem}Notice that a morphism $A\\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is of finite presentation if and only if $B$ is of finite presentation with respect to the category $A\/{\\text{\\bfseries\\sf{Comm}}}({\\tC})$.\n\\end{rem}\n\\begin{rem}\\label{rem:epi}Let ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. We will need to pay special attention to the {\\it epimorphisms} in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$: those morphisms $p:A \\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ such that for all $C \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$, the induced map $\\Hom_{{\\text{\\bfseries\\sf{Comm}}}({\\tC})}(B,C) \\to \\Hom_{{\\text{\\bfseries\\sf{Comm}}}({\\tC})}(A,C)$ is injective. These correspond to monomorphisms in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$. Notice that $p:A \\to B$ is an epimorphism if and only if the multiplication $B\\overline{\\otimes}_{A}B \\to B$ is an isomorphism.\n\\end{rem}\n\\begin{defn}\\label{defn:AbstractFlat}Let ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. An object $V\\in {\\tC}$ is called flat when the functor ${\\tC}\\to{\\tC}$ given by $W \\mapsto V\\overline{\\otimes} W$ is exact (commutes with finite limits).\n\\end{defn}\n\\begin{rem}\\label{rem:SummandOfFlat} Suppose that ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits and that finite products and coproducts agree in ${\\tC}$. Then a finite coproduct of elements of ${\\tC}$ is flat if and only if each of them is individually flat.\n\\end{rem}\n\\begin{defn}\\label{defn:TVf}Let ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. A morphism $p:A \\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is flat when the morphism \\[p^{*}:{\\text{\\bfseries\\sf{Mod}}}(A) \\to {\\text{\\bfseries\\sf{Mod}}}(B)\\] is exact (commutes with finite limits). This precisely says that $B$ is flat in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Mod}}}(A))$.\n\\end{defn}\nSay that $q:\\spec(C) \\to \\spec(A)$ is arbitrary. Consider the Cartesian diagram \n\\begin{equation}\\label{basechange}\\xymatrix{  \\spec(C\\otimes_{A}B) \\ar[r]^{q'} \\ar[d]_{p'} & \\spec(B) \\ar[d]^{p} \\\\\n\\spec(C) \\ar[r]_{q}& \\spec(A). \n}\n\\end{equation}\n\n\n\nUsing the notation of diagram (\\ref{basechange}) There is a natural equivalence \\cite{TV}\n\\begin{equation}\\label{eqn:basechangeequation}p^{*}q_{*} \\Longrightarrow q'_{*}p'^{*}\n\\end{equation}\ncalled base change. \n\n \\begin{defn}A base change of a morphism $p:\\spec(B) \\to \\spec(A)$ is the morphism $p'$ appearing in diagram (\\ref{basechange}) for some $q$.\n\\end{defn}\n\n\\begin{lem}\\label{InvBaseChange}\nLet ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. Suppose that $p:A \\to B$ is a flat morphism in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ then any base change $p'$ of $p$ is also a flat morphism in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$. \n\\end{lem}\n\n{\\bf Proof.}\nBy assumption, $p^{*}: {\\text{\\bfseries\\sf{Mod}}}(A) \\to {\\text{\\bfseries\\sf{Mod}}}(B)$ is exact. Let ${\\tL}$ be a finite category and consider the categories ${\\text{\\bfseries\\sf{Mod}}}(D)^{{\\tL}},$ the category of functors from ${\\tL}$ to ${\\text{\\bfseries\\sf{Mod}}}(D)$. Consider the functor \n\\[\\lim_{{\\tL},D}: {\\text{\\bfseries\\sf{Mod}}}(D)^{{\\tL}} \\to {\\text{\\bfseries\\sf{Mod}}}(D)\n\\]\nwhich takes a diagram in ${\\text{\\bfseries\\sf{Mod}}}(D)$ to its limit.\nConsider the following commutative diagram of natural transformations of functors ${\\text{\\bfseries\\sf{Mod}}}(C)^{{\\tL}} \\to {\\text{\\bfseries\\sf{Mod}}}(B)$: \n\\[\n\\xymatrix{q'_{*}\\lim_{{\\tL},C\\otimes_{A}B} p'^{*} \n\\ar@{=>}@\/_2pc\/[rrrr] \n& \\ar@{=>}[l] \\lim_{{\\tL},B} q'_{*} p'^{*} & \\ar@{=>}[l] \\lim_{{\\tL},B} p^{*}q_{*} \\ar@{=>}[r] & p^{*}q_{*} \\lim_{{\\tL},C} \\ar@{=>}[r] & q'_{*} p'^{*}\\lim_{{\\tL},C}.}\n\\]\n\\\\\nAll the natural transformations except the curved one on the bottom are invertible. This follows from the base change natural equivalences and the fact that the pushforward functors are right adjoints and so commute with limits. Therefore, the natural transformation on the bottom is also invertible.\nSince $q'_{*}$ is conservative, the natural transformation $\\lim_{{\\tL},C\\otimes_{A}B} p'^{*} \\Longrightarrow p'^{*}\\lim_{{\\tL},C}$ of functors ${\\text{\\bfseries\\sf{Mod}}}(C)^{{\\tL}} \\to {\\text{\\bfseries\\sf{Mod}}}(C \\otimes_{A} B)$ is also invertible and so $p'^{*}$ commutes with finite limits. Therefore, $p'$ is flat.\n\\ \\hfill $\\Box$\n\\begin{lem}\\label{lem:left}Let ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. Let $p$ be a morphism in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ and let $p'$ be a base change of $p$.  If $p$ is a monomorphism then so is $p'$.  If $p$ is of finite presentation, so is $p'$.\n\\end{lem}\n{\\bf Proof.} Left to the reader.\n\\ \\hfill $\\Box$\n\nConsider the following very slight modification of Proposition 2.4 and its proof from \\cite{TV} (we only require finite limits and colimits and consider only a special case of their proposition).\n\\begin{lem}\\label{lem:BaseChangeConserv}Let ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. Suppose that a family $\\{p_{i}:X_{i}\\to X\\}$ in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is such that the family $\\{p^{*}_{i}:{\\text{\\bfseries\\sf{Mod}}}(X) \\to {\\text{\\bfseries\\sf{Mod}}}(X_i)\\}$ has a finite conservative subfamily. Then any pull-back family $\\{p_{i}:X_{i}\\times_{X} Y\\to Y\\}$ coming from a base change $Y\\to X$ has the same property.\n\\end{lem}\n{\\bf Proof.} In order to show the base change property, consider $q:Y \\to X$. Choose a finite set $J \\subset I$ such that $\\prod_{i \\in J} p^{*}_{i}$ is conservative. Consider the functor $\\prod_{i \\in J} p^{'*}_{i}$ where $q'_{i}$, $p'_i$ and $p_i$ play the role of $q'$, $p'$ and $p$ in diagram (\\ref{basechange}). In order to show it is conservative, its enough to show that $\\prod_{i \\in J} q'_{i*}p^{'*}_{i}$ is conservative but using equation (\\ref{eqn:basechangeequation}) this is isomorphic to $(\\prod_{i \\in J} p^{*}_{i}) q_{*}$ which is conservative since $q_{*}$ is conservative.\n\\ \\hfill $\\Box$\n\\begin{prop}\\label{TV1}\nLet ${\\tC}$ be a closed symmetric monoidal category with all finite limits and colimits. Consider the families $\\{p_{i}:X_{i}\\to X\\}_{i \\in I}$ in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ such that the family $\\{p^{*}_{i}:{\\text{\\bfseries\\sf{Mod}}}(X) \\to {\\text{\\bfseries\\sf{Mod}}}(X_i)\\}_{i \\in I}$ has a finite conservative subfamily and that each $p_{i}^{*}$ is left exact. These families define a pretopology on ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$. \n\\end{prop}\n{\\bf Proof.} In order to show the base change property, consider $q:Y \\to X$ in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ and let  $q'_{i}$, $p'_i$ and $p_i$ play the role of $q'$, $p'$ and $p$ in diagram (\\ref{basechange}). Lemma \\ref{lem:BaseChangeConserv} implies that the family $\\{p^{*}_{i}\\}$ has a finite conservative subfamily. The fact that the $p_{i}^{*}$ are exact follows from Lemma \\ref{InvBaseChange}.\n\n\\ \\hfill $\\Box$\n\\begin{defn}\\label{defn:fpqc}\nFor any closed symmetric monoidal category ${\\tC}$ which has all finite limits and colimits, the topology coming from Proposition \\ref{TV1} is called the fpqc topology on ${\\text{\\bfseries\\sf{Aff}}}({\\tC})= {\\text{\\bfseries\\sf{Comm}}}({\\tC})^{op}$. When equipped with this topology, we denote this category by ${\\text{\\bfseries\\sf{Aff}}}({\\tC})^{fpqc}.$ The category of sheaves of sets is denoted ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{Aff}}}({\\tC})^{fpqc}).$\n\\end{defn}\n\\begin{defn}\\label{defn:fTVZ}\n The morphism $\\spec(B) \\to \\spec(A)$ is called a formal Zariski open immersion if the corresponding morphism $A \\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is a flat epimorphism (defined in Remark \\ref{rem:epi} and Definition \\ref{defn:TVf}).\n\\end{defn}\n\\begin{defn}\\label{defn:TVZ}\n The morphism $\\spec(B) \\to \\spec(A)$ is called a Zariski open immersion if the corresponding morphism $A \\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is a flat epimorphism of finite presentation (defined in Remark \\ref{rem:epi} and Definitions \\ref{defn:FinitePres} and \\ref{defn:TVf}).\n\\end{defn}\n\\begin{lem}\\label{lem:ConsAgain}If a family $\\{A\\to B_{i}\\}_{i\\in I}$ is conservative and $A'$ is any $A$-algebra then the family $\\{A'\\to B_{i}\\otimes_{A} A'\\}_{i\\in I}$ is conservative.\n\\end{lem}\n{\\bf Proof.}\nThis has already been shown in Proposition \\ref{TV1}.\n\\ \\hfill $\\Box$\n\n\\begin{prop}\\label{prop:formalZarPretop}\nThere is a pretopology whose covering families $\\{A\\to B_{i}\\}_{i\\in I}$ are those families where each $A\\to B_{i}$ is a formal Zariski open immersion and the family $\\{A\\to B_{i}\\}_{i\\in I}$ has a finite conservative subfamily.\n\\end{prop}\n{\\bf Proof.}\nThis follows immediately from Lemmas \\ref{InvBaseChange}, \\ref{lem:ConsAgain} and \\ref{lem:left}.\n\\ \\hfill $\\Box$\n\\begin{defn}\\label{defn:fTV2} The formal Zariski topology on ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is the topology associated to the pretopology from Proposition \\ref{prop:formalZarPretop}. When equipped with this topology, we denote the category by ${\\text{\\bfseries\\sf{Aff}}}({\\tC})^{fZar}$. The category of sheaves of sets is denoted ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{Aff}}}({\\tC})^{fZar}).$\n\\end{defn}\n\\begin{prop}\\label{prop:ZarPretop}\nThere is a pretopology whose covering families $\\{A\\to B_{i}\\}_{i\\in I}$ are those families where each $A\\to B_{i}$ is a Zariski open immersion and the family $\\{A\\to B_{i}\\}_{i\\in I}$ has a finite conservative subfamily.\n\\end{prop}\n{\\bf Proof.}\nThis follows immediately from Lemmas \\ref{InvBaseChange}, \\ref{lem:ConsAgain} and \\ref{lem:left}.\n\\ \\hfill $\\Box$\n\\begin{defn}\\label{defn:TV2} The Zariski topology on ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is the topology associated to the pretopology from Proposition \\ref{prop:ZarPretop}. When equipped with this topology, we denote the category by ${\\text{\\bfseries\\sf{Aff}}}({\\tC})^{Zar}$. The category of sheaves of sets is denoted ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{Aff}}}({\\tC})^{Zar}).$\n\\end{defn}\n\\begin{defn}\\label{defn:hRep} For any affine object $\\spec(A),$ $A \\in {\\text{\\bfseries\\sf{Comm}}}(\\text{\\bfseries\\sf{C}}),$ the presheaf of sets $\\straighth_{A}$ is given by\n\\[{\\text{\\bfseries\\sf{Aff}}}({\\tC}) \\to {\\text{\\bfseries\\sf{Set}}}\n\\]\n\\[\\spec(B) \\mapsto \\Hom_{{\\text{\\bfseries\\sf{Comm}}}(\\text{\\bfseries\\sf{C}})}(A,B).\n\\]\n\\end{defn}\nCor. 2.11 of \\cite{TV} implies that \n\\begin{prop}\\label{TV3}\nFor any $A \\in {\\text{\\bfseries\\sf{Comm}}}(\\text{\\bfseries\\sf{C}})$, the preseheaf $\\straighth_{A}$ is a sheaf for fpqc, the formal Zariski and the Zariski topologies. \n\\end{prop}\n\n\\begin{defn}\\label{defn: ZarOpenSch}\\cite{TV} Let $X \\in {\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$ and $F\\in {\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{Zar})$ be a subsheaf of $X$. Then $F$ is a called a Zariski open of $X$ if there is a family of Zariski opens $\\{X_i \\to X\\}_{i \\in I}$ such that $F$ is the image of the morphism of sheaves $\\coprod_{i \\in I} X_i \\to X$. A morphism $F \\to G$ in ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{Zar})$ is called is a Zariski open immersion if for every $X \\in {\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$ and every $X \\to G$ the induced morphism $F \\times_{G} X \\to X$ is a monomorphism whose image is a Zariski open in $X$.\nThe category ${\\text{\\bfseries\\sf{Sch}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{Zar})$ of schemes is defined to be the full subcategory of ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{Zar})$ of sheaves $F$ such that there exists a family of $X_{i} \\in {\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$ for $i \\in I$ and a morphism $p: \\coprod_{i \\in I} X_i \\to F$ such that $p$ is an epimorphism of sheaves and for each $i$ the morphism $X_i \\to F$ is a Zariski open.\n\\end{defn}\n\n\n\\begin{defn}\\label{defn:GeneralSch} Suppose that ${\\text{\\bfseries\\sf{A}}}$ is a full subcategory of ${\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$ and suppose that $\\tau$ is a subcategory of ${\\text{\\bfseries\\sf{A}}}$ with all objects and such that all morphisms in $\\tau$ are monomorphisms and so that the base change of a morphism in $\\tau$ by an arbitrary morphism of ${\\text{\\bfseries\\sf{A}}}$ is in $\\tau$. Say that $T$ is a pre-topology whose covers consist of families of morphisms where each morphism in the cover belongs to $\\tau$.  Let $X \\in {\\text{\\bfseries\\sf{A}}}$ and $F\\in {\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{A}}}^{T})$ be a subsheaf of $X$. Then $F$ is a called a $\\tau$-open if there is a family of morphisms in $\\tau$ written  $\\{X_i \\to X\\}_{i \\in I}$ such that $F$ is the image of the morphism of sheaves $\\coprod_{i \\in I} X_i \\to X$. A morphism $F \\to G$ in ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{A}}}^{T})$ is called is a $\\tau$-open immersion if for every $X \\in {\\text{\\bfseries\\sf{A}}}$ and every $X \\to G$ the induced morphism $F \\times_{G} X \\to X$ is a monomorphism whose image is a $\\tau$-open in $X$.\nThe category of schemes ${\\text{\\bfseries\\sf{Sch}}}({\\text{\\bfseries\\sf{A}}}, T, \\tau)$ is defined to be the full subcategory of ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{A}}}^{T})$ of sheaves $F$ such that there exists a family of $X_{i} \\in {\\text{\\bfseries\\sf{A}}}$ for $i \\in I$ and a morphism $p: \\coprod_{i \\in I} X_i \\to F$ such that $p$ is an epimorphism of sheaves and for each $i$ the morphism $X_i \\to F$ is a $\\tau$-open immersion.\n\\end{defn}\n\\begin{example}Two interesting general categories of schemes that we have in mind are ${\\text{\\bfseries\\sf{Sch}}}({\\text{\\bfseries\\sf{A}}}, T,\\tau)$ where ${\\text{\\bfseries\\sf{A}}}={\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$. First, the case where $\\tau$ of Zariski open immersions and $T$ is the Zariski pre-topology. Second, the category $\\tau$ of formal Zariski open immersions and $T$ is the formal Zariski pre-topology. \n\\end{example}\n\n\n\\begin{example}\nLet $k$ be any field and consider the category ${\\tC}= {\\text{\\bfseries\\sf{Vect}}}_{k}$. Then ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is the category of $k$-algebras. Recall that a morphism $A \\to B$ of $k$-algebras is of finite presentation in the usual sense when $B$ is finitely generated as an $A$ algebra and the ideal of relations is also finitely generated. Let us temporarily call a morphism TVfp if it satisfies the condition from Definition \\ref{defn:FinitePres}. Consider the functor\n\\[\\straightF : {\\text{\\bfseries\\sf{Set}}} \\to A\/{\\text{\\bfseries\\sf{Comm}}}({\\tC}) \n\\]\nwhich sends each set to the $A$-algebra freely generated by it. Then Lawvere's work on finitary algebraic theories and Corollary 3.13 and the remark following it in \\cite{AR} show that $A \\to B$ is TVfp if and only if there exists finite sets $S_g$ and $S_r$ and an isomorphism in $A\/{\\text{\\bfseries\\sf{Comm}}}({\\tC})$ of the form\n\\[\\colim[\\straightF S_r \\rightrightarrows \\straightF S_g ] \\to B.\n\\]\nSo a morphism $f:A \\to B$ of $k$-algebras is of finite presentation in the categorical sense if and only it is of finite presentation in terms of generators and relations. This fact also appears in the algebraic geometry literature. The implication that finite presentation in terms of generators and relations implies finite presentation in the categorical sense was shown in this case in Lemma III.8.8.2.3 of \\cite{EGA4}. For the opposite implication see \\cite{stacks-project}. \nNow \\cite{EGA4} IV.17.9.1 tells us that a morphism of schemes is a flat monomorphism, locally of finite presentation if and only if it is an open immersion. Since a morphism of affine schemes $\\spec(B) \\to \\spec(A)$ is locally of finite presentation if and only if the corresponding morphism $A \\to B$ realizes $B$ as an $A$-algebra of finite presentation we can conclude that the Zariski open immersions are precisely the standard (Zariski) open immersions in algebraic geometry. The Zariski topology in the sense of relative algebraic geometry agrees with the Zariski topology in the standard sense in the case ${\\tC}= {\\text{\\bfseries\\sf{Vect}}}_{k}$. We should remark that this is the only example in this article for which we can use Lawvere's theory and \\cite{AR} in a straightforward way. Another way to characterize the Zariski open immersion is by replacing the flat epimorphism condition with a homotopy epimorphism condition, i.e. that the natural morphism in the derived category $B\\otimes^{\\mathbb{L}}_{A}B \\to B$ is an isomorphism. It is this condition that we examine this article for the category of Banach spaces, and not the flatness condition which would give a different answer. We comment on the case of vector spaces again in Remark \\ref{rem:recover}.\n\\end{example}\n\\begin{defn}\\label{defn:PresheafZarOpIm} A morphism $f:F \\to G$ in ${\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}}))$ is a Zariski open immersion if for every affine scheme $X$ and every morphism $X \\to G$, the induced morphism $F \\times_{G}X \\to X$ is a monomorphism of presheaves and its image agrees with the image of the map of sheaves $\\coprod_{i \\in I} X_{i } \\to X$ corresponding to a family of Zariski opens $X_{i} \\to X$.  \n\\end{defn}\n\\begin{defn}\\label{defn:AffToPreFlat}\nSuppose  $ F \\in {\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}}))$ and $X \\in {\\text{\\bfseries\\sf{Aff}}}({\\tC})$. A morphism $X \\to F$ is flat if for every $Y \\in {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ and every morphism $Y \\to F$ there is a Zariski open cover $\\coprod Z_{i} \\to X \\times_{F} Y$ such that the combined morphisms $Z_{i} \\to  X \\times_{F} Y \\to Y$ are flat. \n\\end{defn}\n\\begin{defn}\\label{defn:PresheafFlatMor} A morphism $f:F \\to G$ in ${\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}}))$ is flat if for every affine scheme $X$ and every morphism $X \\to G$ and every flat morphism $W \\to X \\times_{G} F$ the composition $W \\to X \\times_{G} F \\to X$ is flat.\n\\end{defn}\n\nConsider the Grothendieck site ${\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{fpqc}$. The category of simplicial objects in ${\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}}))$ is denoted ${\\text{\\bfseries\\sf{SPr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})).$ This category comes with a (local) model structure as explained in \\cite{T}.\n\\begin{defn}\\cite{T} An object $F \\in {\\text{\\bfseries\\sf{SPr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{fpqc})$ is called a pre-Stack. An object $F \\in {\\text{\\bfseries\\sf{SPr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{fpqc})$ is called an fpqc stack if for any $X \\in {\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$ and any hypercovering $H_{\\bullet} \\to X,$ the natural morphism\n\\[F(X) \\to \\text{holim}_{[n] \\in \\Delta} F(H_n)\n\\]\nis an equivalence of simplicial sets. The category ${\\text{\\bfseries\\sf{St}}}({\\text{\\bfseries\\sf{Aff}}}({\\tC})^{fpqc}) =Ho({\\text{\\bfseries\\sf{SPr}}}({\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})^{fpqc}))$ is the category of stacks.\n\\end{defn}\n\nAn equivalent definition to the above is to define pre-stacks as functors ${\\text{\\bfseries\\sf{Aff}}}({\\tC})^{op}\\to \\infty-{\\text{\\bfseries\\sf{Gpd}}}$, where $\\infty-{\\text{\\bfseries\\sf{Gpd}}}$ is the category of infinity groupoids. This is a full subcategory of the category of $(\\infty,1)$-categories for which one could use quasi-categories. $\\infty$-groupoids in this model are Kan simplicial sets. Stacks would be pre-stacks which satisfy descent with respect to hypercovers \\cite{DHI, L1,L2, TVe2,TVe3}. Similarly we could define (pre-)stacks valued in other categories, for instance a pre-stack in categories would be a functor ${\\text{\\bfseries\\sf{Aff}}}({\\tC})^{op} \\to (\\infty,1)-{\\text{\\bfseries\\sf{Cat}}}$. Note that the category of $(1,1)$-categories embeds into the category of $(\\infty,1)$-categories. Using quasi-categories to model $(\\infty,1)$-categories, the nerve of a $1$-category is a quasi-category. We can also view dg-categories as stable quasi-categories tensored over complexes \\cite{Coh}. We will use this later to view categories of quasi-coherent $\\mathcal{O}$-modules, and $\\mathcal{D}$-modules (derived or underived) as pre-stacks valued in categories. There is an inductive definition of an $n$-algebraic stack for $n=0,1,2,\\dots$. An algebraic fpqc stack on ${\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$ is an fpqc stack on ${\\text{\\bfseries\\sf{Aff}}}(\\text{\\bfseries\\sf{C}})$ which is $n$-algebraic for some $n$.\n\n\nOne can study schemes and stacks using with the fpqc or Zariski topology we have defined above. These could be useful in the analytic context as well when ${\\tC}$ is the category of Banach spaces and one can study faithfully flat descent in this context. However, the usual G-topology that is usually studied in non-Archimedean geometry as well as the classical metric topology of complex analytic geometry are finer topologies and have smaller ``open sets\". The localizations from Definition \\ref{defn:AffLocalization} are not flat with respect to the monoidal structure on ${\\text{\\bfseries\\sf{Ban}}}_{k}$ and for this reason we need to introduce new abstractly defined topologies which will fit in well with these facts. To do so, we need to use quasi-abelian categories.\n\n\n\n\\section{Quasi-abelian categories}We review some of Schneiders' theory of quasi-abelian categories. These are special cases of Palamodov's semi-abelian categories and of pseudo-abelian categories. They also have the  structure of a (Quillen) exact category in one natural way.\nThe main reference for this section is \\cite{SchneidersQA}.\n\\begin{defn}\nLet $\\mathcal{E}$ be an additive category with kernels and cokernels. A morphism $f:E\\to F$ is $\\mathcal{E}$ is called strict if the induced morphism \n\\[\\coim(f)\\to \\im(f)\n\\] is an isomorphism. \n\\end{defn}\nHere the image of $f$ is the kernel of the canonical map\n$F \\to \\coker(f)$, and the coimage of $f$ is the cokernel of the canonical map $\\ker (f)\\to E$. \n\n\\begin{defn}\\label{defn:CartCoCart}\nLet $\\mathcal{E}$ be an additive category with kernels and cokernels. We say that $\\mathcal{E}$ is quasi-abelian if it satisfies the following two conditions:\n\\begin{itemize}\n\\item In a cartesian square \n\\begin{equation*}\n\\xymatrix{  E' \\ar[r]^{f'} \\ar[d] & F' \\ar[d] \\\\\nE \\ar[r]_{f}& F}\n\\end{equation*}\n\nIf $f$ is a strict epimorphism then $f'$ is a strict epimorphism.\n\n\\item In a co-cartesian square\n\\begin{equation*}\n\\xymatrix{  E \\ar[r]^{f} \\ar[d] & F \\ar[d] \\\\\nE' \\ar[r]_{f'}& F'}\n\\end{equation*}\nIf $f$ is a strict monomorphism then $f'$ is a strict monomorphism.\n\n\\end{itemize}\n\\end{defn}\n\\begin{rem}\\label{rem:LeftRightInverse} Any morphism in a quasi-abelian category with a right inverse is a strict epimorphism.  Any morphism in a quasi-abelian category with a left inverse is a strict monomorphism. \\end{rem}\n\\begin{defn}\nLet ${\\text{\\bfseries\\sf{E}}}$ be a quasi-abelian category. \nLet $\\xymatrix{ E'\\ar[r]^{e'} & E\\ar[r]^{e''} & E''}$ be a sequence of maps such that $e''\\circ e'=0$. We call such a sequence strictly exact (resp. strictly coexact) if $e'$ (resp. $e''$) is strict and the canonical map \n$\\im(e')\\to \\ker(e'')$ is an isomorphism. A complex \n$E_1\\to \\cdots \\to E_n$ is strictly exact (resp. strictly coexact) if each subsequence $E_{i-1}\\to E_i\\to E_{i+1}$ is strictly exact (resp. strictly coexact).\n\\end{defn}\n\n\\begin{rem}\\label{rem:threetermstrict}\nNote that the sequence \\begin{equation}\\label{equation:threetermstrict}\\xymatrix{ 0\\ar[r] & E\\ar[r]^{u}& F\\ar[r]^{v}& G\\ar[r]& 0}\\end{equation} is strictly exact if and only if $u$ is the kernel of $v$ and $v$ is the cokernel of $u$. Any strict monomorphism or strict epimorphism can be completed to a strictly exact sequence in the form of Equation \\ref{equation:threetermstrict}. \nThis implies that such a sequence is strictly exact if and only if it is strictly coexact.\n\\end{rem}Let ${\\text{\\bfseries\\sf{E}}}$ be a closed symmetric monoidal quasi-abelian category with all finite limits and colimits. Then an object is flat if and only if tensoring with it preserves strict short exact sequences.\n\n\\begin{defn} Call a sequence  $\\xymatrix{ E'\\ar[r]^{e'} & E\\ar[r]^{e''} & E''}$  exact (resp. coexact) if the canonical map \n$\\im(e')\\to \\ker(e'')$ is an isomorphism. A sequence \n$E_1\\to \\cdots \\to E_n$ is exact (resp. coexact) if each subsequence $E_{i-1}\\to E_i\\to E_{i+1}$ is exact (resp. coexact).\n\\end{defn}\n\\begin{rem}\\label{rem:threeterm}\nNote that the sequence \\begin{equation}\\label{equation:threeterm}\\xymatrix{ 0\\ar[r] & E\\ar[r]^{u}& F\\ar[r]^{v}& G\\ar[r]& 0}\\end{equation} is  exact if and only if $\\ker(u)=0$, $\\im(v)=G$ and $\\im(u) \\to \\ker(v)$ is an isomorphism.  Any  monomorphism or epimorphism can be completed to a exact sequence in the form of Equation \\ref{equation:threeterm}. \n\\end{rem}\n\nThe following is remark $1.1.11$ in \\cite{SchneidersQA}:\n\\begin{thm}\nLet ${\\text{\\bfseries\\sf{E}}}$ be a quasi-abelian category. The class of strictly exact short exact sequences endows ${\\text{\\bfseries\\sf{E}}}$ with the structure of an exact category.\n\\end{thm}\n\n\\begin{defn}\nLet ${\\text{\\bfseries\\sf{E}}}$ be a quasi-abelian category. Let $\\straightK({\\text{\\bfseries\\sf{E}}})$ be its homotopy category. The derived category of ${\\text{\\bfseries\\sf{E}}}$ is $\\straightD({\\text{\\bfseries\\sf{E}}})=\\straightK({\\text{\\bfseries\\sf{E}}})\/\\straightN({\\text{\\bfseries\\sf{E}}})$ where $\\straightN({\\text{\\bfseries\\sf{E}}})$ is the full subcategory of strictly exact sequences.\n\\end{defn}\n\n\\begin{defn}\nLet ${\\text{\\bfseries\\sf{E}}}$ be a quasi-abelian category. Let $\\straightK({\\text{\\bfseries\\sf{E}}})$ be its homotopy category. A morphism in \n$\\straightK({\\text{\\bfseries\\sf{E}}})$ is called a strict quasi-isomorphism if its mapping cone is strictly exact. \n\\end{defn}\n\nThe following is 1.2.17, 1.2.19, 1.2.20, 1.2.27 and 1.2.31 in \\cite{SchneidersQA}: \n\\begin{thm}\nLet ${\\text{\\bfseries\\sf{E}}}$ be a quasi-abelian category.\n\\begin{enumerate}\n\\item $\\straightD({\\text{\\bfseries\\sf{E}}})$ has a canonical t-structure (the left t-structure). A complex $E$ belongs to $D^{\\leq 0}$ if and only if it is strictly exact in strictly positive degrees. $E$ belongs to $D^{\\geq 0}$ if and only if it is strictly exact in strictly negative degrees.\n\\item The heart of this t-structure $\\LH({\\text{\\bfseries\\sf{E}}})$, is equivalent to the localization of the full subcategory of $\\straightK({\\text{\\bfseries\\sf{E}}})$ consisting of complexes E of the form \n\\begin{equation}\n\\xymatrix{ 0\\ar[r] & E\\ar[r]^{u}& F\\ar[r]& 0}\n\\end{equation}\nwhere $u$ is a monomorphism and $F$ is in degree $0$, by the multiplicative system formed by morphisms which are both cartesian and cocartesian. \n\\item There is a canonical fully faithful functor $\\sI:{\\text{\\bfseries\\sf{E}}}\\to LE({\\text{\\bfseries\\sf{E}}})$. A sequence $E'\\to E\\to E''$ is strictly exact in ${\\text{\\bfseries\\sf{E}}}$ if and only if $\\sI(E')\\to \\sI(E)\\to \\sI(E'')$ is exact in $\\LH({\\text{\\bfseries\\sf{E}}})$.\n\\item The functor $I$ induces an equivalence between $\\straightD({\\text{\\bfseries\\sf{E}}})$ and $\\straightD(\\LH({\\text{\\bfseries\\sf{E}}}))$. This equivalence sends the (left) t-structure on $\\straightD({\\text{\\bfseries\\sf{E}}})$ to the standard t-structure on $\\straightD(\\LH({\\text{\\bfseries\\sf{E}}}))$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{rem}\nThe embedding $\\sI:{\\text{\\bfseries\\sf{E}}}\\to \\LH({\\text{\\bfseries\\sf{E}}})$ is universal \nin the sense that induces an equivalence for any abelian category $\\mathcal{F}$ between left strictly exact functors from ${\\text{\\bfseries\\sf{E}}}$ to $\\mathcal{F}$ and left exact functors \nfrom $\\LH({\\text{\\bfseries\\sf{E}}})$ to $\\mathcal{F}$. In this sense $\\LH({\\text{\\bfseries\\sf{E}}})$ is the (left) abelian envelope of ${\\text{\\bfseries\\sf{E}}}$. See 1.2.33 in \\cite{SchneidersQA}.\n\\end{rem}\n\n\\begin{defn}\\label{defn:Projective}\nLet ${\\text{\\bfseries\\sf{E}}}$ be an additive category with kernels and cokernels. An object $I$ is called injective (resp. strongly injective) if the functor $E\\mapsto \\Hom(E,I)$ is exact (resp. strongly exact), i.e. for any strict (resp. arbitrary) monomorphism $u:E\\to F$, the induced map $\\Hom(F,I)\\to \\Hom(E,I)$ is surjective. Dually, $P$ is called projective (resp. strongly projective) if the functor $E\\mapsto \\Hom(P,E)$ is exact (resp. strongly exact), i.e. for any strict (resp. arbitrary) epimorphism $u:E\\to F$, the associated map $\\Hom(P,E)\\to \\Hom(P,F)$ is surjective.     \n\\end{defn}\n\n\\begin{defn}\\label{defn:enough}\nA quasi-abelian category ${\\text{\\bfseries\\sf{E}}}$ has enough projectives if for any object $E$ there is a strict epimorphism $P\\to E$ where $P$ is projective. A quasi-abelian category ${\\text{\\bfseries\\sf{E}}}$ has enough injectives if for any object $E$ there is a strict monomorphism $E \\to I$ where $I$ is injective.\n\\end{defn}\n\nThe following is 1.3.24 in \\cite{SchneidersQA}:\n\\begin{lem}\nLet ${\\text{\\bfseries\\sf{E}}}$ be a quasi-abelian category. \n\\begin{enumerate}\n\\item An object $P$ of ${\\text{\\bfseries\\sf{E}}}$ is projective if and only if $\\sI(P)$ is projective in $\\LH({\\text{\\bfseries\\sf{E}}})$.\n\\item ${\\text{\\bfseries\\sf{E}}}$ has enough projectives if and only if $\\LH({\\text{\\bfseries\\sf{E}}})$ has enough projectives. In this case an object of $\\LH({\\text{\\bfseries\\sf{E}}})$ is projective if it is isomorphic  to $\\sI(P)$ where $P$ is projective in ${\\text{\\bfseries\\sf{E}}}$.\n\\end{enumerate}\n\\end{lem}\n\nThe following is 1.3.22 in \\cite{SchneidersQA}:\n\\begin{thm}\nLet ${\\text{\\bfseries\\sf{E}}}$ be a quasi-abelian category with enough projectives (resp. injectives). Let $\\text{\\bfseries\\sf{P}}$ be the full additive subcategory of projective objects (resp. $\\text{\\bfseries\\sf{I}}$ the category of injective objects). The canonical functor $\\sK^-(\\text{\\bfseries\\sf{P}})\\to \\sD^-({\\text{\\bfseries\\sf{E}}})$ (resp. $\\sK^+(\\text{\\bfseries\\sf{I}})\\to \\sD^+({\\text{\\bfseries\\sf{E}}})$) is an equivalence.\n\\end{thm}\n\n\n\n\n\\subsection{Closed symmetric monoidal quasi-abelian categories.}\n\nThe following is 1.5.1 in \\cite{SchneidersQA}:\n\\begin{prop}\\label{prop:StrictIFF}\nSuppose that ${\\tC}$ is a closed symmetric monoidal quasi-abelian category with all finite limits and colimits. Suppose that $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$.  The category ${\\text{\\bfseries\\sf{Mod}}}(A)$ \nis quasi-abelian and the forgetful functor ${\\text{\\bfseries\\sf{Mod}}}(A)\\to {\\tC}$ preserves limits and colimits. A morphism in ${\\text{\\bfseries\\sf{Mod}}}(A)$ is strict if and only if it is strict in ${\\tC}$. \n\\end{prop}\n\n\n\\begin{lem}Suppose that ${\\tC}$ is a closed symmetric monoidal quasi-abelian category with all finite limits and colimits. Using Remark \\ref{rem:LeftRightInverse} we see that for any $V  \\in {\\text{\\bfseries\\sf{Mod}}}(A)$ the canonical morphism\n\\[V \\overline{\\otimes} A \\to V\n\\]\nis a strict epimorphism and the canonical morphism\n\\[V \\to \\underline{\\Hom}(A,V)\n\\]\nis a strict monomorphism.\n\\end{lem}\n\n\n\\begin{defn}\\label{defn:finiteAmodAbstract}\nSuppose that ${\\tC}$ is a closed symmetric monoidal quasi-abelian category with all finite limits and colimits. An object $V$ is called finite if there is a strict epimorphism $\\coprod_{i=1}^{n} \\text{id}_{{\\tC}} \\to V$ in ${\\tC}$ for some finite non-negative integer $n$. In the case that ${\\tC}={\\text{\\bfseries\\sf{Mod}}}(A)$ for $A$ a commutative monoid in a closed symmetric monoidal quasi-abelian category, we denote the full subcategory of finite objects by ${\\text{\\bfseries\\sf{Mod}}}^{f}(A)$.\n\\end{defn}\n\n\\begin{lem}\\label{lem:FreeProjCofreeInj} Suppose that $0 \\to L \\to M \\to N \\to 0$ is a strictly exact sequence in ${\\text{\\bfseries\\sf{Mod}}}(A)$ and $F =A\\overline{\\otimes} P \\in {\\text{\\bfseries\\sf{Mod}}}(A)$ is free and $C=\\underline{\\Hom}(A,I) \\in {\\text{\\bfseries\\sf{Mod}}}(A)$ is cofree. Then the sequences \n\\[0 \\to \\Hom_{A}(F,L) \\to  \\Hom_{A}(F,M)\\to  \\Hom_{A}(F,N) \\to 0\n\\]\nand\n\\[0 \\to \\Hom_{A}(N,C) \\to  \\Hom_{A}(M,C)\\to  \\Hom_{A}(L,C) \\to 0\n\\]\nare exact. If the sequence $0 \\to L \\to M \\to N \\to 0$ is only exact and $F$ is strictly free and $C$ is strictly cofree we can make the same conclusion.\n\\end{lem}\n{\\bf Proof.}\nUsing Lemma \\ref{lem:MeyerProperties} these sequences are isomorphic to the sequences\n\\[0 \\to \\Hom(P,L) \\to  \\Hom(P,M)\\to  \\Hom(P,N) \\to 0\n\\]\nand \n\\[0 \\to \\Hom(N,I) \\to  \\Hom(M,I)\\to  \\Hom(L,I) \\to 0\n\\]\nwhich are exact by definition of projectivity and injectivity (or the strict versions).\n\\hfill $\\Box$\n\\begin{lem}\\label{lem:MaintainProjInj}If $P$ is projective in ${\\tC}$ then $P\\overline{\\otimes} A$ is projective in ${\\text{\\bfseries\\sf{Mod}}}(A)$. Similarly, if $I$ is injective in ${\\tC}$ then $\\underline{\\Hom}(A,I)$ is injective in ${\\text{\\bfseries\\sf{Mod}}}(A)$.\n\\end{lem}\n{\\bf Proof.}\nBoth of these facts are immediately implied by Lemma \\ref{lem:FreeProjCofreeInj} together with Remark \\ref{rem:threetermstrict}.\n\\hfill $\\Box$\n\\begin{lem} \\label{lem:MonoEpiPreserved} The functor $E \\mapsto E\\overline{\\otimes} A$ takes epimorphisms in ${\\tC}$ to epimorphisms in ${\\text{\\bfseries\\sf{Mod}}}(A)$. The functor $E \\mapsto \\underline{\\Hom}(A,E)$ takes monomorphisms in ${\\tC}$ to monomorphisms in ${\\text{\\bfseries\\sf{Mod}}}(A)$.\n\\end{lem}\n{\\bf Proof.}\nThis follows directly from the definitions and Lemma \\ref{lem:MeyerProperties}.\n\\hfill $\\Box$\n\n\\begin{defn}\\label{defn:KerFlat} For $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$ a module $M$ in ${\\text{\\bfseries\\sf{Mod}}}(A)$ is called kernel flat if for any morphism $f:E \\to F$ in ${\\text{\\bfseries\\sf{Mod}}}(A)$  the natural morphism \n\\begin{equation}\\label{equation:want4flatAbstr}B\\overline{\\otimes} \\ker(f) \\to \\ker(f_{M})\n\\end{equation}\nis an isomorphism where $f_{M}$ is defined as $\\text{id}_{M} \\overline{\\otimes}_{A}f:M \\overline{\\otimes}_{A}E \\to M \\overline{\\otimes}_{A}F.$ A morphism $A \\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is called kernel flat if it makes $B$ kernel flat over $A$.\n\\end{defn}\n\n\\begin{lem}\\label{lem:FlatkerFlat}Suppose that ${\\tC}$ is a closed symmetric monoidal quasi-abelian category.  An object $V\\in {\\tC}$ is kernel flat if and only it is flat (see Definition \\ref{defn:AbstractFlat}). Therefore, a morphism of algebras $p: A \\to B$ is kernel flat (see Definition \\ref{defn:KerFlat}) if and only if it is flat (Definition \\ref{defn:AbstractFlat}) in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$. \n\n\\end{lem}\n\n{\\bf Proof.}\nFirst of all if $p$ is flat it is clearly kernel flat since a kernel is a type of limit. In the other direction suppose that $p$ is kernel flat. It means that tensoring with $B$ commutes with kernels. Note that every limit over a finite diagram can be written as a combination of finite products and kernels. Finite products are isomorphic to finite coproducts and the functor given by tensoring with $B$ commutes with coproducts and hence it commutes with finite products. Therefore, tensoring with $B$ commutes with finite limits and hence $p$ is flat. \n\n\\ \\hfill $\\Box$\n\\begin{lem}\\label{lem:HelpWithEnough} Suppose that ${\\tC}$ is a closed symmetric monoidal quasi-abelian category with all finite limits and colimits. Suppose that $A \\in {\\text{\\bfseries\\sf{Comm}}}({\\tC})$. If the category ${\\tC}$ has enough projectives \nthen the category ${\\text{\\bfseries\\sf{Mod}}} (A)$ has enough projectives. If the category ${\\tC}$ has enough injectives \nthen the category ${\\text{\\bfseries\\sf{Mod}}}(A)$ has enough injectives. \n\\end{lem}\n{\\bf Proof.} Suppose that ${\\tC}$ has enough projectives. Suppose that $V \\in {\\text{\\bfseries\\sf{Mod}}}(A)$. Choose a strict epimorphism in ${\\tC}$ of the form $P \\to V$ where $P$ is projective in ${\\tC}$.  Lemma \\ref{lem:MaintainProjInj} implies that $P\\overline{\\otimes} A$ is projective in ${\\text{\\bfseries\\sf{Mod}}}(A).$ Consider the morphism $P\\overline{\\otimes} A \\to V.$ We need to show it is a strict epimorphism in ${\\text{\\bfseries\\sf{Mod}}}(A).$ It factorizes as \n\\begin{equation}\nP \\overline{\\otimes} A \\to V \\overline{\\otimes} A  \\to V.\n\\end{equation}\nThe second morphism is a strict epimorphism because it admits a right inverse. The arrow $P \\overline{\\otimes} A \\to V \\overline{\\otimes} A$  is an epimorphism by Lemma \\ref{lem:MonoEpiPreserved} and in fact a strict epimorphism because the monoidal product with $A$ is a left adjoint functor and preserves cokernels. Therefore ${\\text{\\bfseries\\sf{Mod}}}(A)$ has enough projectives.\nSuppose that ${\\tC}$ has enough injectives. Choose a strict monomorphism in ${\\tC}$ of the form $V \\to I$ where $I$ is injective in ${\\tC}$. Lemma \\ref{lem:MaintainProjInj} implies that $\\underline{\\Hom}(A,I)$ is injective in ${\\text{\\bfseries\\sf{Mod}}}(A).$ Consider the morphism $V \\to \\underline{\\Hom}(A,I).$ We need to show that it is a strict monomorphism in ${\\text{\\bfseries\\sf{Mod}}}(A).$ It factorizes as \\begin{equation}\nV \\to \\underline{\\Hom}(A,V) \\to \\underline{\\Hom}(A,I).\n\\end{equation}\nNotice that here, we are considering $\\underline{\\Hom}(A,V)$ and $\\underline{\\Hom}(A,I)$ as elements of \n${\\text{\\bfseries\\sf{Mod}}}(A)$ using the action of $A$ on itself. The first arrow is a strict monomorphism because it admits a left inverse. Using Lemma \\ref{lem:MonoEpiPreserved}, $\\underline{\\Hom}(A,V) \\to \\underline{\\Hom}(A,I)$ is a monomorphism in ${\\text{\\bfseries\\sf{Mod}}}(A)$ and in fact a strict monomorphism because the internal Hom from $A$ is a right adjoint functor and preserves kernels. Therefore ${\\text{\\bfseries\\sf{Mod}}}(A)$ has enough injectives.\n\\hfill $\\Box$\n\\begin{lem}\\label{lem:BaseChangeInj}Suppose that $\\sF: {\\tC} \\to \\text{\\bfseries\\sf{D}}$ is a functor which has a right adjoint $\\sR$ and which \npreserves strict monomorphisms (preserves monomorphisms). Then an injective (strongly injective) in $\\text{\\bfseries\\sf{D}}$ is an injective (strongly injective) when considered in ${\\tC}$ via $\\sR$ .\n\\end{lem}\n{\\bf Proof.}\n\nSuppose that $I \\in \\text{\\bfseries\\sf{D}}$ is injective (strongly injective). Then consider a strict monomorphism (monomorphism) $E\\to F$ in ${\\tC}.$ Then $\\sF(E) \\to \\sF(F)$ is a strict monomorphism (monomorphism) in ${\\tC}.$ We have a commutative diagram\n\\[\\xymatrix{{\\tC}(F,\\sR (I))  \\ar[r]& {\\tC} (E,\\sR (I)) \\\\\n\\text{\\bfseries\\sf{D}}(\\sF (F),I) \\ar[r] \\ar[u]& \\text{\\bfseries\\sf{D}} (\\sF (E),I). \\ar[u]\n}\n\\]\nBecause the upwards arrows are isomorphisms and the lower horizontal arrow is surjective, the upper horizontal arrow is surjective as well. Therefore, $I$ is  injective (strongly injective) when considered as an object of ${\\tC}$. \n\\hfill $\\Box$\n\n\\begin{lem}\\label{lem:BaseChangeProj}Suppose that $\\sG: {\\tC} \\to \\text{\\bfseries\\sf{D}}$ is a functor which has a left adjoint $\\sL$ and which preserves strict epimorphisms (preserves epimorphisms). Then a projective (strongly projective) in $\\text{\\bfseries\\sf{D}}$  is projective (strongly projective) when considered in ${\\tC}$.\n\\end{lem}\n{\\bf Proof.}\nSuppose that $P \\in \\text{\\bfseries\\sf{D}}$ is projective (strongly projective). Consider a strict epimorphism (epimorphism) $E\\to F$ in $\\text{\\bfseries\\sf{D}}(A).$ Then by $\\sG(E) \\to \\sG(F)$ is a strict epimorphism (epimorphism) in $\\text{\\bfseries\\sf{D}}.$ We have a commutative diagram\n\\[\\xymatrix{{\\tC}(\\sL (P),E)  \\ar[r]& {\\tC} (\\sL (P),F)\\\\\n\\text{\\bfseries\\sf{D}}(P,\\sG (E))\\ar[u] \\ar[r]& \\text{\\bfseries\\sf{D}}(P,\\sG (F)) \\ar[u].\n}\n\\]\nBecause the upwards arrows are isomorphisms and the lower horizontal arrow is surjective, the upper horizontal arrow is surjective as well. Therefore, $P$ is projective (strongly projective) when considered in ${\\tC}$. \n\\ \\hfill $\\Box$\n\nLet ${\\text{\\bfseries\\sf{E}}}$ be a closed symmetric monoidal quasi-abelian category and let $A\\in {\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{E}}})$. \nThe following is contained in 2.1.18 in \\cite{SchneidersQA}: If $P$ is projective in ${\\text{\\bfseries\\sf{E}}}$ then $A\\overline{\\otimes} P$ is projective in ${\\text{\\bfseries\\sf{Mod}}}(A)$.\n\n\n\n \n\\subsection{Derived Functors}\nLet $\\sF: {\\tC} \\to {\\tD}$ be an additive functor betwen quasi-abelian categories ${\\tC}$ and ${\\tD}$. Schneiders gave the following definitions in 1.3.2 of \\cite{SchneidersQA}\n\\begin{defn} A full additive subcategory ${\\tP}$ of ${\\tC}$ is called $\\sF$-projective if:\n\\begin{enumerate}\n\\item for any object $V$ of ${\\tC}$ there is an object $P$ of ${\\tP}$ and a strict epimorphism $P\\to V$\n\\item in any strictly exact sequence \n\\[0 \\to V' \\to V \\to V'' \\to 0\n\\]\nof ${\\tC}$ where $V$ and $V''$ are objects of ${\\tP}$, $V'$ is as well\n\\item for any strictly exact sequence \n\\[0 \\to V' \\to V \\to V'' \\to 0\n\\]\nof ${\\tC}$ where $V, V'$ and $V''$ are objects of ${\\tP}$, the sequence \n\\[0 \\to \\sF(V') \\to \\sF(V) \\to \\sF(V'') \\to 0\n\\]\nis strictly exact in ${\\tD}.$\n\\end{enumerate}\n A full additive subcategory ${\\tI}$ of ${\\tC}$ is called $\\sF$-injective if:\n\\begin{enumerate}\n\\item for any object $V$ of ${\\tC}$ there is an object $I$ of ${\\tI}$ and a strict monomorphism $V\\to I$\n\\item in any strictly exact sequence \n\\[0 \\to V' \\to V \\to V'' \\to 0\n\\]\nof ${\\tC}$ where $V$ and $V''$ are objects of ${\\tI}$, $V'$ is as well\n\\item for any strictly exact sequence \n\\[0 \\to V' \\to V \\to V'' \\to 0\n\\]\nof ${\\tC}$ where $V, V'$ and $V''$ are objects of ${\\tI}$, the sequence \n\\[0 \\to \\sF(V') \\to \\sF(V) \\to \\sF(V'') \\to 0\n\\]\nis strictly exact in ${\\tD}.$\n\\end{enumerate}\n\\end{defn}\nSchneiders also includes the following (Lemma 1.3.3 \\cite{SchneidersQA})\n\\begin{lem}\nLet ${\\tC}$ be a quasi-abelian category and let ${\\tP}$ be a subset of the objects of ${\\tC}$. Assume that for any object $V$ of ${\\tC}$ there is a strict epimorphism $P \\to V$ with $P \\in {\\tP}$. Then for each object $V$ of $C^{-}({\\tC})$ there is a quasi-isomorphism $u:P \\to V$ with $P$ in $C^{-}({\\tP})$ and such that each $u^{k}:P^{k} \\to V^{k}$ is a strict epimorphism.\n\\end{lem}\n\nFrom this we get (proposition 1.3.5 \\cite{SchneidersQA}):\n\\begin{prop}\\label{lem:derivable}\nLet $\\sF: {\\tC} \\to {\\tD}$ be an additive functor between quasi-abelian categories ${\\tC}$ and ${\\tD}$.\n\\begin{enumerate}\n\\item Assume that ${\\tC}$ has an $\\sF$-projective subcategory. Then $\\sF$ has a left derived functor $L\\sF:D^-({\\tC})\\to D^-({\\tD})$.\n\\item Assume that ${\\tC}$ has an $\\sF$-injective subcategory. Then $\\sF$ has a right derived functor $R\\sF:D^+({\\tC})\\to D^+({\\tD})$.\n\\end{enumerate}\n\\end{prop}\n\\begin{defn}In the situations of Lemma \\ref{lem:derivable}, $\\sF$ is called explicitly left derivable or explicitly right derivable.\n\\end{defn}\nHere, derived functors are defined as usual by their universal property.\n\n\\begin{rem}\\label{rem:sse2sse}Note that if $\\sF$ is exact (sends strict short exact sequences to strict short exact sequences) then the full subcategory ${\\tC}$ itself is an $\\sF$-projective (and injective) subcategory. Hence exact functors are always derivable.   \n\\end{rem}\n\n\n\nAs in the abelian case, projective and injectives form $\\sF$-projective and $\\sF$-injective subcategories (remark 1.3.21 \\cite{SchneidersQA}):\n\\begin{prop}\\label{prop:ProjFProj}\nLet $\\sF: {\\tC} \\to {\\tD}$ be an additive functor between quasi-abelian categories ${\\tC}$ and ${\\tD}$. \n\\begin{enumerate}\n\\item Assume that ${\\tC}$ has enough projectives. Then the full subcategory of projective objects is a $\\sF$-projective subcategory and therefore can be used to explicitly left derive the functor $\\sF.$\n\\item Assume that ${\\tC}$ has enough injectives. Then the full subcategory of injective objects is a $\\sF$-injective subcategory  and therefore can be used to explicitly right derive the functor $\\sF.$\n\\end{enumerate}\n\\end{prop}\nWe also have the following (remark 1.3.7 \\cite{SchneidersQA}):\n\\begin{lem}\\label{lem:acyclic}\nLet $\\sF: {\\tC} \\to {\\tD}$ be an additive functor between quasi-abelian categories ${\\tC}$ and ${\\tD}$. Assume that $\\sF$ has a right derived functor $R\\sF:D^+({\\tC})\\to D^+({\\tD})$. Call an object $I$ $\\sF$-acyclic if $R\\sF(I)\\cong \\sF(I)$. Assume that for any object \n$A$, there is an $\\sF$-acyclic object $I$ and a monomorphism \n$A\\to I$. Then the $\\sF$-acyclic objects form a $\\sF$-injective subcategory.  Assume that $\\sF$ has a left derived functor $L\\sF:D^-({\\tC})\\to D^-({\\tD})$. Call an object $P$ $\\sF$-acyclic if $L\\sF(P)\\cong \\sF(P)$. Assume that for any object \n$A$, there is an $\\sF$-acyclic object $P$ and a epimorphism \n$P \\to A$. Then the $\\sF$-acyclic objects form a $\\sF$-projective subcategory. \n\\end{lem}\n\\begin{defn}Let ${\\tC}$ be a closed symmetric monoidal quasi-abelian category with monoidal structure $\\overline{\\otimes}$. An object $V$ of ${\\tC}$ is called $\\overline{\\otimes}$-acyclic if $V$ is $\\sF$-acyclic for all of the functors $\\sF:{\\tC} \\to {\\tC}$ given by $U \\mapsto U \\overline{\\otimes} W$ for any object $W$ in ${\\tC}$.\n\\end{defn}\n\\subsection{Topologies based on homological algebra}\nUsing the homological algebra in this section, we now introduce some more classes of morphisms and Grothendieck topologies on a closed symmetric monoidal quasi-abelian category $\\text{\\bfseries\\sf{C}}$ with all finite limits and colimits.\n\\begin{lem}For any morphism $p:\\spec(B)\\to \\spec(A)$ in ${\\text{\\bfseries\\sf{Aff}}}({\\tC}),$ the induced morphism $p_{*}:{\\text{\\bfseries\\sf{Mod}}}(B)\\to {\\text{\\bfseries\\sf{Mod}}}(A)$ is derivable to a functor $\\sD^{-}(B) \\to \\sD^{-}(A).$ \n\\end{lem}\n{\\bf Proof.} This functor sends strict exact sequences to strict exact sequences so this follows from Remark \\ref{rem:sse2sse}.\n\\ \\hfill $\\Box$\n\\begin{defn}\\label{defn:homotopyEpi}A morphism $p:\\spec(B)\\to \\spec(A)$ in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is called a homotopy monomorphism in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ if the induced morphism $p_{*}:\\sD^{-}(B) \\to \\sD^{-}(A)$ is fully faithful.\n\\end{defn}\nNotice that by considering $i=0$ in Definition \\ref{defn:homotopyEpi} we see that a homotopy epimorphism in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is in particular an epimorphism in ${\\text{\\bfseries\\sf{Comm}}}({\\tC}).$\n\n\\begin{lem}\\label{lem:ComposHom}The composition of homotopy monomorphisms in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is a homotopy monomorphism in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$.\n\\end{lem}\n{\\bf Proof.}\nThis follows from the fact that the composition of fully faithful functors is fully faithful.\n\\ \\hfill $\\Box$\n\\begin{lem}\n\nAssume that $p:\\spec(B)\\to \\spec(A)$ in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ and that the functor $p^{*}:{\\text{\\bfseries\\sf{Mod}}}(A) \\to {\\text{\\bfseries\\sf{Mod}}}(B)$ given by tensoring with $B$ over $A$ is explicitly left derivable to a functor $\\mathbb{L}p^{*}:\\sD^{-}(A)\\to \\sD^{-}(B)$. Then $p$ is homotopy monomorphism if and only if the natural morphism of functors $\\mathbb{L}p^{*}p_{*}\\to  id_{\\sD^{-}(B)}$ is an isomorphism.\n\n\\end{lem}\n{\\bf Proof.}\nWe have natural isomorphisms for any objects $M,N \\in \\sD^{-}(B)$ \n\\[\\Hom_{\\sD^{-}(B)}(\\mathbb{L}p^{*}p_{*}M,N)   \\cong \\Hom_{\\sD^{-}(A)}(p_{*}M,p_{*}N) .\n\\]\nTherefore, if $\\mathbb{L}p^{*}p_{*}\\to  id_{\\sD^{-}(B)}$ is a isomorphism then $p$ is a homotopy epimorphism. The converse follows from a simple application of the Yoneda lemma.\n\\ \\hfill $\\Box$\n\\begin{lem}\\label{lem:HomotopyMon}\n Assume that $p:\\spec(B)\\to \\spec(A)$ is a morphism in ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ and that the functor ${\\text{\\bfseries\\sf{Mod}}}(A) \\to {\\text{\\bfseries\\sf{Mod}}}(B)$ given by tensoring with $B$ over $A$ is explicilty left derivable to a functor $\\sD^{-}(A)\\to \\sD^{-}(B)$. Then $p$ is homotopy monomorphism if and only if $B\\overline{\\otimes}^{\\mathbb{L}}_{A}B\\cong B$.\n\\end{lem}\n{\\bf Proof.} \nFor any object $M$ of $\\sD^{-}(B)$ we have \n\\begin{equation*}\nM\\overline{\\otimes}^{\\mathbb{L}}_{A}B\\cong M\\overline{\\otimes}^{\\mathbb{L}}_{B}(B\\overline{\\otimes}^{\\mathbb{L}}_{A}B).\n\\end{equation*}\n\nHence $\\mathbb{L}p^{*}p_{*}\\to  id_{\\sD^{-}(B)}$ is an isomorphism if and only if  we have natural isomorphisms $M\\overline{\\otimes}^{\\mathbb{L}}_{A}B\\cong M$ for any $M \\in \\sD^{-}(B)$ which happens if and only if $B\\overline{\\otimes}^{\\mathbb{L}}_{A}B\\cong B$.\n\\ \\hfill $\\Box$\n\n\n\\begin{defn}\\label{defn:fhTVZ} Let ${\\tC}$ be a closed, symmetric monoidal quasi-abelian category with enough projectives.\n The morphism $\\spec(B) \\to \\spec(A)$ of ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is called a homotopy formal Zariski open immersion if the corresponding morphism $A \\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is a homotopy epimorphism.\n\\end{defn}\n\\begin{defn}\\label{defn:hTVZ} Let ${\\tC}$ be a closed, symmetric monoidal quasi-abelian category with enough projectives.\n The morphism $\\spec(B) \\to \\spec(A)$ of ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ is called a homotopy Zariski open immersion if the corresponding morphism $A \\to B$ in ${\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is a homotopy epimorphism of finite presentation.\n\\end{defn}\n\n\\begin{defn}\\label{defn:Amitsur}The Amitsur complex of a morphism $f:A \\to B$ in $ {\\text{\\bfseries\\sf{Comm}}}({\\tC})$ is the complex $\\mathscr{A}(f)$ given by \n\\[0 \\to A \\to B \\to B\\overline{\\otimes}_{A}B \\to  B\\overline{\\otimes}_{A}B\\overline{\\otimes}_{A}B \\to \\cdots\n\\]\nwhere the morphism $B^{\\overline{\\otimes}_{A}^{m}}\\to B^{\\overline{\\otimes}_{A}^{m+1}}$ is defined by \n\\[d(b_{1} \\overline{\\otimes} b_{2} \\overline{\\otimes} \\cdots \\overline{\\otimes} b_{m})= \\sum_{i=1}^{m+1}(-1)^{i}b_{1} \\overline{\\otimes} \\cdots  \\overline{\\otimes} b_{i-1}\\otimes 1 \\overline{\\otimes} b_{i} \\overline{\\otimes} \\cdots \\overline{\\otimes} b_{m}.\n\\]\nIn the case that we can chose a decomposition $f= \\prod_{i=1}^{n} f_{i}:A \\to \\prod_{i=1}^{n} B_{i}=B$ there is a strictly included subcomplex $\\mathscr{A}^{a}(f)\\to \\mathscr{A}(f)$ where $\\mathscr{A}^{a}(f)$ is given by the finite complex \n\\[0 \\to A \\to \\prod_{1 \\leq i_1 \\leq n} B_{i}  \\to \\prod_{1\\leq i_1<i_2\\leq n} B_{i_1}\\overline{\\otimes}_{A} B_{i_2} \n\\to \\cdots \\to B_{1} \\overline{\\otimes}_{A} \\cdots \\overline{\\otimes}_{A} B_{n} \\to 0\\]\nwith the induced differentials. \n\\end{defn}\n\\begin{defn}\\label{defn:hZt} Let ${\\tC}$ be a closed, symmetric monoidal quasi-abelian category with enough projectives. We call a full subcategory ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ \\emph{homotopy Zariski transversal} if it is closed under fiber products and for any homotopy monomorphism $\\spec (B) \\to \\spec (A)$ in ${\\text{\\bfseries\\sf{A}}}$ and for any morphism $\\spec (C) \\to \\spec (A)$ in ${\\text{\\bfseries\\sf{A}}}$ \n\\[\\text{the natural morphism} \\ \\ B \\overline{\\otimes}_{A}^{\\mathbb{L}}C \\to  B \\overline{\\otimes}_{A}C \\ \\ \\text{is an isomorphism.}\n\\]\n\\end{defn}\n\\begin{lem}\\label{lem:IfThen} If ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ homotopy Zariski transversal subcategory then the base change of a homotopy monomorphism is a homotopy monomorphism.\n\\end{lem}\n{\\bf Proof.} If $A \\to B$ is a homotopy epimorphism in ${\\text{\\bfseries\\sf{A}}}$ and $A \\to C$ is any morphism in ${\\text{\\bfseries\\sf{A}}}$ then \n\\[(B \\overline{\\otimes}_{A} C) \\overline{\\otimes}^{\\mathbb{L}}_{C} (B \\overline{\\otimes}_{A} C) \\cong (B \\overline{\\otimes}^{\\mathbb{L}}_{A} C) \\overline{\\otimes}^{\\mathbb{L}}_{C} (B \\overline{\\otimes}^{\\mathbb{L}}_{A} C) \\cong B \\overline{\\otimes}^{\\mathbb{L}}_{A} (B \\overline{\\otimes}^{\\mathbb{L}}_{A} C)\\cong (B \\overline{\\otimes}^{\\mathbb{L}}_{A} B) \\overline{\\otimes}^{\\mathbb{L}}_{A} C \\cong B \\overline{\\otimes}^{\\mathbb{L}}_{A} C\\cong B \\overline{\\otimes}_{A} C.\n\\]\n\\ \\hfill $\\Box$\n\nThe following definition comes from work \\cite{RR} of Ramis-Ruget on quasi-coherent sheaves in complex analytic geometry. Based on their work, quasi-coherent modules in the complex analytic context were discussed in the book \\cite{EP}. Our definition is inspired by that one. \n\\begin{defn}\\label{defn:RRqc}Let ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ be a homotopy Zariski transversal subcategory. For $\\spec(A) \\in {\\text{\\bfseries\\sf{A}}}$ define ${\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{A}}}}^{RR}(A)$ to be the full subcategory of ${\\text{\\bfseries\\sf{Mod}}}(A)$ consisting of modules $M$ such that $M$ is transversal to all homotopy epimorphisms in ${\\text{\\bfseries\\sf{A}}}$. That is, we consider modules $M$ such that the natural morphism \\[M \\overline{\\otimes}_{A}^{\\mathbb{L}}B \\to  M \\overline{\\otimes}_{A}B \\] is an isomorphism for all homotopy epimorphisms $A \\to B$. We call these RR-quasi-coherent modules.\n\\end{defn}\n\\begin{lem}Let ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ be a homotopy Zariski transversal subcategory. If $\\spec(C) \\to \\spec(A)$ is any morphism in ${\\text{\\bfseries\\sf{A}}}$ then the morphism $A \\to C$ gives $C$ the structure of an object of ${\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{A}}}}^{RR}(A)$.\n\\end{lem}\n{\\bf Proof.} This is an obvious consequence of Definition \\ref{defn:hZt}.\n\\ \\hfill $\\Box$\n\\begin{lem}\\label{lem:RRpreserve} If ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ homotopy Zariski transversal subcategory then pushforwards by morphisms in ${\\text{\\bfseries\\sf{A}}}$ preserve the category of RR-quasi-coherent modules. If a morphism is transverse to a RR-quasi-coherent module, then the pull-back of this module by that morphism is also RR-quasi-coherent. Therefore, pullbacks by homotopy monomorphisms or flat morphisms in ${\\text{\\bfseries\\sf{A}}}$ also preserve this category. \n\\end{lem}\n{\\bf Proof.} Say we are given a morphism $ \\spec(C) \\to \\spec(A)$ in  ${\\text{\\bfseries\\sf{A}}}$. Given any object $M$ of ${\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{A}}}}^{RR}(C)$ and a homotopy monomorphism $\\spec(B) \\to \\spec(A)$, we have \n\\[M \\overline{\\otimes}^{\\mathbb{L}}_{A} B \\cong M \\overline{\\otimes}^{\\mathbb{L}}_{C} (C \\overline{\\otimes}_{A}^{\\mathbb{L}}B) \\cong M \\overline{\\otimes}^{\\mathbb{L}}_{C} (C \\overline{\\otimes}_{A}B) \\cong M \\overline{\\otimes}_{C} (C \\overline{\\otimes}_{A}B) \\cong M \\overline{\\otimes}_{A} B.\n\\]\nThis proves the statement about push-forwards. For the statements about pullbacks, fix a morphism $\\spec(D) \\to \\spec(C)$ in ${\\text{\\bfseries\\sf{A}}}$ (which we will pullback with) which is transverse to $M$ and a homotopy monomorphism $\\spec(E) \\to \\spec(D)$ in ${\\text{\\bfseries\\sf{A}}}$. Then\n\\[(M \\overline{\\otimes}_{C} D)\\overline{\\otimes}^{\\mathbb{L}}_{D}E \\cong (M \\overline{\\otimes}^{\\mathbb{L}}_{C} D)\\overline{\\otimes}^{\\mathbb{L}}_{D}E \\cong M \\overline{\\otimes}^{\\mathbb{L}}_{C} E \\cong M \\overline{\\otimes}_{C} E \\cong (M \\overline{\\otimes}_{C} D)\\overline{\\otimes}_{D}E.\n\\]\n\\ \\hfill $\\Box$\n\nAs a corollary, we can get a helpful analogue of Lemma \\ref{lem:BaseChangeConserv}: \n\\begin{cor}\\label{cor:BaseChangeConservHzAR}Let ${\\tC}$ be a closed symmetric monoidal quasi-abelian category. Suppose that ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ homotopy Zariski transversal subcategory. Suppose that a family $\\{p_{i}:X_{i}\\to X\\}$ in ${\\text{\\bfseries\\sf{A}}}$ of homotopy monomorphisms is such that the family $\\{p^{*}_{i}:{\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{A}}}}^{RR}(X) \\to {\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{A}}}}^{RR}(X_i)\\}$ has a finite conservative subfamily. Then any pull-back family $\\{p_{i}:X_{i}\\times_{X} Y\\to Y\\}$ coming from a base change $Y\\to X$ has the same property.\n\\end{cor}\n{\\bf Proof.} The proof of Lemma \\ref{lem:BaseChangeConserv} uses only the functors $q_{*}$, $q'_{*}$, $p_{i}^{*}$ and ${p'}_{i}^{*}$. The first three preserve the categories of RR-quasi-coherent modules by Lemma \\ref{lem:RRpreserve} and the last one  does by Lemmas \\ref{lem:IfThen} and \\ref{lem:RRpreserve}. Therefore, that same proof works here.\n\\ \\hfill $\\Box$\n\n\\begin{prop}\\label{hfTV1}Let ${\\tC}$ be a closed, symmetric monoidal quasi-abelian category with enough projectives. Let ${\\text{\\bfseries\\sf{A}}}$ be a homotopy transversal subcategory of ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$.  Consider the families $\\{p_{i}:X_{i}\\to X\\}_{i \\in I}$ in ${\\text{\\bfseries\\sf{A}}}$ such that the family $\\{p^{*}_{i}:{\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{A}}}}^{RR}(X) \\to {\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{A}}}}^{RR}(X_i)\\}_{i \\in I}$ has a finite conservative subfamily and that each $p_{i}$ is a homotopy monomorphism. These families define a pretopology on ${\\text{\\bfseries\\sf{A}}}$. \n\\end{prop}\n{\\bf Proof.} In order to show the base change property, consider $q:Y \\to X$ in ${\\text{\\bfseries\\sf{A}}}$ and let  $q'_{i}$, $p'_i$ and $p_i$ play the role of $q'$, $p'$ and $p$ in diagram (\\ref{basechange}). Lemma \\ref{lem:BaseChangeConserv} implies that the family $\\{p^{*}_{i}\\}$ has a finite conservative subfamily. The fact that the $p'_{i}$ are homotopy monomorphisms follows from the assumption.\n\n\\ \\hfill $\\Box$\n\nLet ${\\tC}$ be a closed symmetric monoidal quasi-abelain category with enough projectives. Let ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ be a homotopy Zariski transversal subcategory. Say that $q:\\spec(C) \\to \\spec(A)$ is arbitrary. Say we are given $A, B, C \\in {\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{sC}}})$.  Consider a Cartesian diagram \n\\begin{equation}\\label{Derivedbasechange}\\xymatrix{  \\spec(C\\otimes^{\\mathbb{L}}_{A}B) \\ar[r]^{q'} \\ar[d]_{p'} & \\spec(B) \\ar[d]^{p} \\\\\n\\spec(C) \\ar[r]_{q}& \\spec(A). \n}\n\\end{equation}\n\n\n\nUsing the notation of diagram (\\ref{Derivedbasechange}) there is a natural equivalence \n\\begin{equation}\\label{eqn:basechangeequation}(\\mathbb{L}p^{*})q_{*} \\Longrightarrow q'_{*}(\\mathbb{L}p'^{*})\n\\end{equation}\ncalled base change. \n\n\\begin{lem}\\label{lem:DerivedBaseChangeConserv}Let ${\\tC}$ be a closed symmetric monoidal quasi-abelain category with enough projectives. Let ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$ be a homotopy Zariski transversal subcategory. Suppose that a family $\\{p_{i}:X_{i}\\to X\\}$ of homotopy monomorphisms in ${\\text{\\bfseries\\sf{A}}}$ is such that the family $\\{\\mathbb{L}p^{*}_{i}:D^{-}(X) \\to D^{-}(X_i)\\}$ has a finite conservative subfamily. Then any pull-back family $\\{p_{i}:X_{i}\\times_{X} Y\\to Y\\}$ coming from a base change $Y\\to X$ in ${\\text{\\bfseries\\sf{A}}}$ has the same property.\n\\end{lem}\n{\\bf Proof.}  In order to show the base change property, consider $q:Y \\to X$. Choose a finite set $J \\subset I$ such that $\\prod_{i \\in J} \\mathbb{L}p^{*}_{i}$ is conservative. Consider the functor $\\prod_{i \\in J} \\mathbb{L}{p'}^{*}_{i}$ where $q'_{i}$, $p'_i$ and $p_i$ play the role of $q'$, $p'$ and $p$ in diagram (\\ref{basechange}). In order to show it is conservative, its enough to show that $\\prod_{i \\in J} q'_{i*}\\mathbb{L}{p'}^{*}_{i}$ is conservative. By the definition of homotopy Zariski transversal, the natural morphism $X_{i}\\times_{X} Y \\to  X_{i}\\times^{h}_{X} Y$ is an isomorphism. Therefore, we can use  (\\ref{eqn:basechangeequation}) to conclude that $\\prod_{i \\in J} q'_{i*}\\mathbb{L}{p'}^{*}_{i}$ is natrually equivalent to $(\\prod_{i \\in J} \\mathbb{L}p^{*}_{i}) q_{*}$ which is conservative since $q_{*}$ is conservative.\n\\ \\hfill $\\Box$\n\\begin{rem}Notice that the {\\it derived} base change of a homotopy monomorphism is always a homotopy monomorphism but there is no reason that such a thing would work for ordinary base changes. In order to construct a Gronthendieck topology, we need the admissible opens to be preserved by base change. Because the current article is not about derived geometry which will be discussed in \\cite{BeKr2}, we defined our Gronthendieck topology on a homotopy transversal subcategory ${\\text{\\bfseries\\sf{A}}} \\subset {\\text{\\bfseries\\sf{Aff}}}({\\tC})$. The main application in this article is the case where ${\\tC}$ is the category ${\\text{\\bfseries\\sf{Ban}}}_{k}$ for a non-Archimedean valued field $k$ and ${\\text{\\bfseries\\sf{A}}}$ is the opposite to the category of affinoid algebras. In the preprints \\cite{BaBe} and \\cite{BaBeKr} we take ${\\tC}$ to be the category of Ind-Banach spaces and ${\\text{\\bfseries\\sf{A}}}$ to be categories of dagger affinoid and Stein algebras which we define. In these upcoming articles we do not assume the field is non-Archimedean so they apply to complex analytic geometry as well. In the article on derived analytic geometry \\cite{BeKr2}, we take ${\\tC}$ to be the monoidal model category of simplicial Ind-Ban spaces.\n\\end{rem}\n\n\\begin{defn}\\label{defn:fhZar}Let ${\\tC}$ be a closed, symmetric monoidal quasi-abelian category with enough projectives.  Let ${\\text{\\bfseries\\sf{A}}}$ be homotopy Zariski transversal subcategory of ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$ . The topology coming from Proposition \\ref{hfTV1} is called the formal homotopy Zariski topology on ${\\text{\\bfseries\\sf{A}}}$. When equipped with this topology, we denote this category by ${\\text{\\bfseries\\sf{A}}}^{fhZar}.$ The category of sheaves of sets is denoted ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{A}}}^{fhZar}).$ The category of schemes is denoted by ${\\text{\\bfseries\\sf{Sch}}}({\\text{\\bfseries\\sf{A}}}^{fhZar}).$\n\\end{defn}\n\n\\begin{prop}\\label{hTV1}\nConsider the families $\\{p_{i}:X_{i}\\to X\\}_{i \\in I}$ in ${\\text{\\bfseries\\sf{A}}}$ such that the family $\\{p^{*}_{i}:{\\text{\\bfseries\\sf{Mod}}}^{RR}(X) \\to {\\text{\\bfseries\\sf{Mod}}}^{RR}(X_i)\\}_{i \\in I}$ has a finite conservative subfamily and that each $p_{i}$ is a homotopy monomorphism of finite presentation. These families define a pretopology on ${\\text{\\bfseries\\sf{A}}}$. \n\\end{prop}\n{\\bf Proof.} In order to show the base change property, consider $q:Y \\to X$ in ${\\text{\\bfseries\\sf{A}}}$ and let  $q'_{i}$, $p'_i$ and $p_i$ play the role of $q'$, $p'$ and $p$ in diagram (\\ref{basechange}). Lemma \\ref{lem:BaseChangeConserv} implies that the family $\\{p^{*}_{i}\\}$ has a finite conservative subfamily. The fact that the $p'_{i}$ are homotopy monomorphisms follows from the assumption.\n\n\\ \\hfill $\\Box$\n\n\\begin{defn}\\label{defn:hZar}Let ${\\tC}$ be a closed, symmetric monoidal quasi-abelian category with enough projectives. Let ${\\text{\\bfseries\\sf{A}}}$ be homotopy Zariski transversal subcategory of ${\\text{\\bfseries\\sf{Aff}}}({\\tC})$. The topology coming from Proposition \\ref{hTV1} is called the homotopy Zariski topology on ${\\text{\\bfseries\\sf{A}}}$. When equipped with this topology, we denote this category by ${\\text{\\bfseries\\sf{A}}}^{hZar}.$ The category of sheaves of sets is denoted ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{A}}}^{hZar}).$ The category of schemes is denoted by ${\\text{\\bfseries\\sf{Sch}}}({\\text{\\bfseries\\sf{A}}}^{hZar}).$\n\\end{defn}\n\n\n\n\n\n\\section{Main Theorems}\n\\subsection{From Berkovich geometry to Banach algebraic geometry}\n Let $k$ be a non-Archimedean valuation field. \nWe now introduce the full subcategory ${\\text{\\bfseries\\sf{Afnd}}}_{k} \\subset {\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$. The objects in this category are the $k$-affinoid algebras from the literature on Berkovich analytic spaces.\n\\begin{defn}\\label{defn:NiceAlg} For any finite ordered set of positive real numbers $r=(r_1, \\dots, r_n)$, let \\[k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\} \\in {\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})\\] be the completion of $k[x_{1}, \\dots,  x_{n}]$ with respect to the norm \n\\[\\|\\sum_{I \\in \\mathbb{Z}^{n}_{\\geq 0}} a_{I} x^{I}\\|_{r}=\\text{max}_{I}\\{|a_{I}|r^{I}\\}\n\\] \nwhere for each $I= (i_1, \\dots, i_n) \\in \\mathbb{Z}^{n}_{\\geq 0}$,  $a_{I}= a_{i_1, \\dots, i_n}$ and $x^{I}=x^{i_1}_{1} \\cdots x^{i_n}_{n}$. \n\\end{defn}\n\\begin{lem} More concretely, we have\n\\begin{equation}\\label{equation:concrete}\nk \\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\} = \\{ \\sum_{} a_{I}x^{I} \\in k[[x_1, \\dots, x_n]] \\ \\ | \\ \\ \\lim_{|I| \\to \\infty} |a_{I}| r^{I} = 0 \\}\n\\end{equation}\nwhere $|I|= i_{1} + \\cdots +i_{n}$ and we equip the right hand side with the norm \n\\[\\|\\sum_{I \\in \\mathbb{Z}^{n}_{\\geq 0}} a_{I} x^{I}\\|_{r}=\\sup_{I}\\{|a_{I}|r^{I}\\}\n\\]\n\\end{lem}\n{\\bf Proof.} First, notice that the right hand side is complete: given a Cauchy sequence $\\{f^{(j)}\\}$ in the right hand side of (\\ref{equation:concrete}), note that for each $I$, the sequence of coefficients $f^{(j)}_{I}$ of $x^I$ must be Cauchy, and hence converge in $k$, to some $f_{I}.$ Define $f=\\sum_{I \\in \\mathbb{Z}^{n}_{\\geq 0}} f_{I} x^{I}.$ It is easy to check that $f$ lies in the right hand side using the inequality\n\\[|f_{I}|r^{I} \\leq \\max \\{|f_{I} - f^{(j)}_{I}|r^{I},|f^{(j)}_{I}|r^{I} \\}.\n\\]\n \nIf $f$ is an element of the right hand side, let $f^{(j)} = \\sum_{I \\in \\mathbb{Z}^{n}_{\\geq 0}, |I| \\leq j} f_{I} x^{I}.$ These polynomials $f^{(j)}$ converge to $f$ so the polynomials are dense in the right hand side.\n\\ \\hfill $\\Box$\n\\begin{rem}Notice by Lemma \\ref{lem:sumsConverge} that the right hand side consists precisely of the formal power series which can be evaluated on elements $(x_1, \\dots, x_n) \\in k^{n}$ such that $|x_{i}| \\leq r_i$ for $i=1, \\dots, n$.\n\\end{rem}\n\n\\begin{defn} A $k$-affinoid algebra $\\mathcal{A}$ is an object in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$ which admits an admissible surjection \n\\[k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\} \\to \\mathcal{A}.\n\\]\nwhose multiplication is contracting (see Definition \\ref{defn:nonexpanding}). The full category of ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$ consisting of such objects is denoted ${\\text{\\bfseries\\sf{Afnd}}}_{k}$.\n\\end{defn}\n Notice that the completed symmetric algebra in ${\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k}$ on $k_{r_1} \\oplus k_{r_2} \\oplus \\cdots \\oplus k_{r_n}$ is just $k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\}$ from Definition \\ref{defn:NiceAlg}. So these algebras are precisely the free commutative monoids in ${\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k}$. In fact, we have\n\\begin{lem}Affinoid algebras are precisely the finitely presented objects in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k})$. This means that they are exactly those objects which are the cokernel of a morphism in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k})$ of free commutative algebra objects: \n\\[k\\{s_{1}^{-1}y_{1}, \\dots, s_{m}^{-1} y_{m}\\} \\to k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\}\n\\]\n\\end{lem}\n{\\bf Proof.} Let $\\mathcal{A} = k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\}\/I$ be an affinoid algebra. It is known \\cite{Te} that $k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\}$ is Noetherian. Therefore, $I$ must be finitely generated by some finite set of non-zero elements $f_1, \\dots, f_m \\in k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\}$. Let $s_{j} = \\|f_{j}\\|$ for $j= 1, \\dots, m$. Consider the morphism of algebras\n\\[k\\{s_{1}^{-1}y_{1}, \\dots, s_{m}^{-1} y_{m}\\} \\to k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\}\n\\]\ndetermined by the (contracting) morphism of Banach spaces \n\\[k_{s_1} \\oplus \\cdots \\oplus k_{s_m} \\to k\\{r_{1}^{-1}x_{1}, \\dots, r_{n}^{-1} x_{n}\\}\n\\]\ndetermined by the maps sending $1 \\in k_{s_j}$ to $f_j$. Its image is the ideal generated by $f_1, \\dots, f_m$.\n\n\\ \\hfill $\\Box$\n\n\n\\begin{defn}\\label{defn:AffLocalization} A $k$-affinoid localization is a morphism $\\mathcal{A} \\to \\mathcal{D}$ of $k$-affinoid algebras such that the morphism $|\\mathcal{M}(\\mathcal{D})|\\to |\\mathcal{M}(\\mathcal{A})|$ is injective, the image of $|\\mathcal{M}(\\mathcal{D})|$ is closed in $|\\mathcal{M}(\\mathcal{A})|$ and any morphism of $k$-affinoid algebras $\\mathcal{A} \\to \\mathcal{B}$ such that $|\\mathcal{M}(\\mathcal{B})|$ lands in the image of  $|\\mathcal{M}(\\mathcal{D})|$ factors as $\\mathcal{A} \\to \\mathcal{D} \\to \\mathcal{B}$. \n\\end{defn}\nNote that $k$-affinoid localizations correspond to the subspaces known as affinoid domain embeddings. They are usually written as $\\mathcal{A} \\to \\mathcal{A}_{V}$ where $V$ is the closed image and they satisfy the property (proven in 2.2.2 (iv) of \\cite{Ber1990}) that \n\\[\\mathcal{A}_{V_1 \\cap V_2} \\cong \\mathcal{A}_{V_1} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_2} \n\\]\nfor affinoid domains $V_1$ and $V_2$.\n\nThree important examples are the localizations corresponding to the rational, Weirstrass and Laurent domains. Any affinoid domain is a union of a finite collection of rational domains.\n\n\\begin{defn}\\label{defn:RatDom}A rational localization is a morphism of affinoid algebras of the form \n\\[\\mathcal{A} \\to \\mathcal{A} \\{\\frac{T_1}{r_1}, \\dots, \\frac{T_m}{r_m} \\}\/(g T_1 - f_1, \\dots, gT_m - f_m)\n\\]\nfor some $g, f_j \\in \\mathcal{A}$ such that $(f_1,\\dots, f_m,g)=1$.\n\\end{defn}\nNotice that the norm of this morphism is one. \n\n\\begin{rem}\n\nProposition 2.2.4 (ii) of \\cite{Ber1990} tells us that $\\mathcal{A}_{V}$ is a flat $\\mathcal{A}$-algebra with respect to the algebraic (non-completed) tensor product, however it is not flat with respect to $\\widehat{\\otimes}_{\\mathcal{A}}$. Note that since completions commute with colimits, a rational localization $\\mathcal{A}_{V}$ is the completion of \n$\\mathcal{A}[T_1, \\dots, T_n]\/(g T_1 - f_1, \\dots, gT_n - f_n)$ with respect to the residue semi-norm coming from the semi-norm in Definition \\ref{def:NDiskrelA}. We have an \nisomorphism of $\\mathcal{A}$ modules \n\\[\\mathcal{A}_g= \\mathcal{A}[S]\/(g S-1) \\to \\mathcal{A}[T_1, \\dots, T_n]\/(g T_1 - f_1, \\dots, gT_m - f_m)\n\\]\ngiven by $S = b+\\sum_{i=1}^{n}a_i T_i$ where $bg + \\sum_{i=1}^{n}a_if_i = 1$. The inverse is given by $T_i=f_iS$. Therefore $\\mathcal{A}[T_1, \\dots, T_n]\/(g T_1 - f_1, \\dots, gT_n - f_n)$ is flat over $\\mathcal{A}$ in the algebraic sense. Given any $\\mathcal{M} \\in {\\text{\\bfseries\\sf{Mod}}}(\\mathcal{A})$ we have a morphism $\\mathcal{M}_{g} \\to \\mathcal{M}\\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_V$ and $\\mathcal{M}\\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_V$ is the completion of $\\mathcal{M}_{g}=\\mathcal{M}[T_1, \\dots, T_n]\/(g T_1 - f_1, \\dots, gT_m - f_m)$ with the residue semi-norm of the algebraic tensor product $\\mathcal{M}[T_1, \\dots, T_n]$.\n\\end{rem}\n\\begin{defn}\\label{defn:WDom}A Weirstrass localization is a rational localization for which $g=1$.\n\\end{defn}\n\\begin{defn}\\label{defn:LDom}A Laurent loaclization is a localization of the form \n\\[\\mathcal{A}\\to \\mathcal{A}\\{\\frac{T_1}{p_1}, \\dots, \\frac{T_n}{p_n}, q_1S_1, \\dots, q_mS_m\\}(T_1 - f_1, \\dots, T_n-f_n, g_1S_1-1, \\dots, g_mS_m-1)\n\\]\nwhere the $p_i$ and $q_j$ are positive reals and $f_i,g_j \\in \\mathcal{A}$.\n\\end{defn}\n\\begin{defn}\\label{defn:CAcomplex}\nFix a system $\\mathcal{A} \\to \\mathcal{A}_{V_i}$ which give a cover of $\\mathcal{M}(\\mathcal{A})$ by a finite collection of affinoid domains $V_i$. The \\v{C}ech-Amitsur complex is the complex \n\\begin{equation}\\label{equation:CA} 0 \\to \\mathcal{M} \\to \\prod_{i} \\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i} \\to \\prod_{i,j}\\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_j} \\to \\cdots.\n\\end{equation}\nTo any morphism $\\mathcal{M} \\to \\mathcal{N}$ we have the obvious morphism of \\v{C}ech-Amitsur complexes. \nThe complex written here differs from the standard long exact sequence defined in section 8.2 of \\cite{BGR} in that we consider the completed tensor products everywhere whereas they only complete the tensor products between the various $\\mathcal{A}_{V_i}$ terms. The standard complex mentioned is exact as proven in the Acyclicity Theorem (Proposition 2.2.5 of \\cite{Ber1990}). The above complex is not always exact, however.\n\\end{defn}\n\\begin{lem}\\label{lem:CAcplxISexact} The complex in definition \\ref{defn:CAcomplex} is strictly exact for modules $\\mathcal{M}$ such that the natural morphism \n\\[\\mathcal{M}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\prod_{i}\\mathcal{A}_{V_i}\\to \\mathcal{M}\\widehat{\\otimes}_{\\mathcal{A}}\\prod_{i}\\mathcal{A}_{V_i}\n\\] is an isomorphism.\n\\end{lem}\n\n{\\bf Proof.}\n\nThis follows from an easy double complex argument using free resolutions.\n\\ \\hfill $\\Box$\n\n\n\\begin{lem}\\label{lem:easyLW} Let $\\mathcal{A}$ be an affinoid algebra over a non-Archimedean  valuation field $k$. Let $\\mathcal{A}_{V}$ be the localization $\\mathcal{A}_{V}= \\mathcal{A}\\{\\frac{T}{s}\\}\/(T-f)$ for some $f\\in \\mathcal{A}$ or $\\mathcal{A}_{V}= \\mathcal{A}\\{\\frac{T}{s}\\}\/(gT-1)$ for some $g\\in \\mathcal{A}$. Let $\\mathcal{B}$ be an affinoid $\\mathcal{A}$-algebra. Then the natural morphism\n\\[\\mathcal{B} \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{A}_{V} \\to   \\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V}\\] is an isomorphism in $\\sD^{-}(\\mathcal{A})$. In particular,  by taking $\\mathcal{B}=\\mathcal{A}_{V}$ we see that the morphisms $\\mathcal{A}\\to \\mathcal{A}\\{\\frac{T}{s}\\}\/(gT-1)$ and  the morphism $\\mathcal{A}\\to \\mathcal{A}\\{\\frac{T}{s}\\}\/(T-f)$ homotopy epimorphisms.\n\\end{lem}\n{\\bf Proof.} \nNotice that it enough to show that for any affinoid algebra $\\mathcal{C}$ and any element $f\\in \\mathcal{C}$ the morphism \n\\begin{equation}\\label{mono1}\\mathcal{C}\\{\\frac{T}{s}\\} \\stackrel{T-f}\\to \\mathcal{C}\\{\\frac{T}{s}\\}\n\\end{equation}\nis a strict monomorphism and for any $g \\in \\mathcal{C}$ the morphism\n\\begin{equation}\\label{mono2}\\mathcal{C}\\{\\frac{T}{s}\\} \\stackrel{gT-1}\\to \\mathcal{C}\\{\\frac{T}{s}\\}\n\\end{equation}\nis a strict monomorphism. Indeed, we can use in the case $\\mathcal{C}=\\mathcal{A}$\n\\[\\mathcal{A}\\{\\frac{T}{s}\\} \\stackrel{T-f}\\to \\mathcal{A}\\{\\frac{T}{s}\\}\n\\]\nor\n\\[\\mathcal{A}\\{\\frac{T}{s}\\} \\stackrel{gT-1}\\to \\mathcal{A}\\{\\frac{T}{s}\\}\n\\]\nas resolutions of $\\mathcal{A}_{V}$ whose terms are projective and $\\widehat{\\otimes}_{\\mathcal{A}}$-acyclic. Then by taking the completed tensor product over $\\mathcal{A}$ with $\\mathcal{B}$ we find a representative for $\\mathcal{B} \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{A}_{V}$ which looks like \n\\[\\mathcal{B}\\{\\frac{T}{s}\\} \\stackrel{T-f}\\to \\mathcal{B}\\{\\frac{T}{s}\\}\n\\]\nor\n\\[\\mathcal{B}\\{\\frac{T}{s}\\} \\stackrel{gT-1}\\to \\mathcal{B}\\{\\frac{T}{s}\\}\n\\]\nwhere the $f$ and $g$ here are the images of the original ones in $\\mathcal{B}$. These complexes are immediately recognized as also being special cases of (\\ref{mono1}) and (\\ref{mono2}) in the case $\\mathcal{C}=\\mathcal{B}.$ This shows that the operation of taking the completed tensor product over $\\mathcal{A}$ with $\\mathcal{B}$ is (strictly) exact.\n\nWe start by showing that (\\ref{mono1}) and (\\ref{mono2}) are monomorphisms which using Lemma \\ref{lem:BanProps} simply means to show that they are injective. Multiplication by $gT-1$ is clearly injective as can be seen by looking at the lowest order term in $T$. Consider the ascending sequence of ideals of $\\mathcal{C}$ given by the kernel of the morphisms of $\\mathcal{C}$ to itself defined by multiplication by $f^{i}$: \\[\\ker(f) \\subset \\ker(f^{2}) \\subset \\ker(f^{3}) \\subset \\cdots. \\] Since $\\mathcal{C}$ is Noetherian by Proposition 2.1.3 of \\cite{Ber1990} this sequence must terminate at some $\\ker(f^{N}).$ Suppose that \\[(T-f)\\left(\\sum_{j=0}^{\\infty} a_{j} T^{j} \\right)= 0\\] with $\\sum_{j=0}^{\\infty} a_{j} T^{j} \\neq 0$ and let $a_{i}$ be the first non-zero coeficient. Then $f^{N}a_{i+N}= a_{i}$ and so $f^{N+1}a_{i+N}= fa_{i}=0$ and so $a_{i+N} \\in \\ker(f^{N+1})= \\ker(f^{N}).$ This shows that $a_{i}=0,$ a contradiction. Therefore, multiplication by $T-f$ is injective.\nLet us assume for a moment that $k$ is non-trivially valued. To show that these morphisms are strict one must simply show that the set-theoretic image is closed. However, this set-theoretic image in both cases is an ideal and all ideals in affinoid algebras are closed by Proposition 2.1.3 of \\cite{Ber1990}. Therefore, they are strict monomorphisms in this case. Now if $k$ is arbitrary we find that these two morphisms become strict after tensoring with the non-trivially valued field introduced in Proposition 2.1.2 of \\cite{Ber1990}. Therefore, by this same proposition, they were strict monomorphisms all along.\n\\ \\hfill $\\Box$\n\n\n\\begin{lem}\\label{lem:LCSrHoEpis}Let $\\mathcal{A}_{V}$ be a rational localization of an affinoid $k$-algebra $\\mathcal{A}$. Let $\\mathcal{B}$ be an affinoid $k$-algebra. Then the natural morphism\n\\[\\mathcal{A}_{V}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B} \\to \\mathcal{A}_{V}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B} \\] \nis an isomorphism in $\\sD^{-}(\\mathcal{A})$. In particular taking $\\mathcal{B}=\\mathcal{A}_{V}$, any rational, Weirstrass, or Laurent localization $\\mathcal{A}\\to \\mathcal{A}_{V}$ is a homotopy epimorphism.\n\\end{lem}\n\n{\\bf Proof.}\n\nAssume now that $k$ is non-trivially valued and consider the rational localization \n\\[\\mathcal{A}_{V} = \\mathcal{A} \\{\\frac{T_1}{r_1}, \\dots, \\frac{T_m}{r_m} \\}\/(g T_1 - f_1, \\dots, gT_m - f_m)\n\\]\nwhere $r_{i}>0$ and $(f_1, \\dots, f_m,g)=1.$\nNotice that following Proposition 1 of 7.2.4 of \\cite{BGR} $|g|$ cannot be arbitrarily small for $|\\;| \\in \\mathcal{M}(\\mathcal{A}_{V})$ and in fact we can realize the rational localization as an Laurent localization of a Weierstrass localization. That is, there is an $\\epsilon>0$ such that\n\\[\\mathcal{A}_{V}\n\\cong \\left(\\mathcal{A}\\{ \\frac{S}{\\epsilon^{-1}}\\}\/(gS-1)\\right)\\{\\frac{T_1}{r_1}, \\dots, \\frac{T_m}{r_m} \\}\/( T_1 - \\frac{f'_1}{g'}, \\dots, T_m -\\frac{f'_m}{g'})\n\\]\nwhere the $f'_i$ and $g'$ are the image of $f_i$ and $g$ in $\\mathcal{A}\\{\\epsilon S\\}\/(gS-1).$ This corresponds to the fact that we can choose $\\epsilon >0$ such that the conditions $|f_{i}|\\leq r_{i}|g|$ and $(f_1, \\dots, f_m,g)=1$ are equivalent to $|g|>\\epsilon$ and $|\\frac{f_{i}}{g}|\\leq r_{i}$. Therefore Lemma \\ref{lem:easyLW} implies that the natural morphism\n\\[\\mathcal{A}_{V}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B} \\to \\mathcal{A}_{V}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B} \\] \nis an isomorphism in $\\sD^{-}(\\mathcal{A})$. \nNow if $k$ is trivially valued, consider, using Proposition 2.1.2 of \\cite{Ber1990}, a field $K_r$  containing $k$ which is flat over $k$ with respect to the completed tensor product and non-trivially valued. When we take the tensor product of the canonical morphism \\[\\mathcal{A}_{V}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B}\\to \\mathcal{A}_{V}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B}\\] with $K_{r}$ over $k$ we get, using the flatness of $K_r$, that\n\\[(\\mathcal{A}_{V} \\widehat{\\otimes}_{k} K_r)\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}  \\widehat{\\otimes}_{k} K_r}(\\mathcal{B} \\widehat{\\otimes}_{k} K_r) \\to \\mathcal{A}_{V} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B} \\widehat{\\otimes}_{k} K_r\n\\] is an isomorphism in $\\sD^{-}(\\mathcal{A}\\widehat{\\otimes}_{k}K_r)$ and hence the original morphism $\\mathcal{A}_{V}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V}\\to \\mathcal{A}_{V} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B}$ is an isomorphism in $\\sD^{-}(\\mathcal{A}).$\n\n\\ \\hfill $\\Box$\n\\begin{lem}Let $\\mathcal{A}_{W_1}$ and $\\mathcal{A}_{W_2}$ be affinoid localizations of an affinoid $k$-algebra $\\mathcal{A}$ corresponding to subdomains $W_1$ and $W_2$. Assume also that $W_1 \\cup W_2$ is an affinoid subdomain. Let $\\mathcal{B}$ be an affinoid $k$-algebra. Assume that the morphisms \n\\[\\mathcal{A}_{W_i}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B} \\to \\mathcal{A}_{W_i}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B}\n\\]\nare isomorphisms for $i=1,2$. Assume also that the morphism\n\\[\\mathcal{A}_{W_1 \\cap W_2}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B} \\to \\mathcal{A}_{W_1 \\cap W_2}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B}\n\\]\nis an isomorphism. Then the morphism \n\\[\\mathcal{A}_{W_1 \\cup W_2}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B} \\to \\mathcal{A}_{W_1 \\cup W_2}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B}\n\\]\nis an isomorphism.\n\\end{lem}\n{\\bf Proof.}\nThis follows immediately from considering the strict short exact sequence \n\\[0 \\to \\mathcal{A}_{W_1 \\cup W_2} \\to \\mathcal{A}_{W_1} \\times \\mathcal{A}_{W_2} \\to \\mathcal{A}_{W_1 \\cap W_2} \\to 0\n.\\]\n\n\\ \\hfill $\\Box$\nFinally, we are able to show in the following theorem that affinoid subdomains of affinoids give examples homotopy monomorphisms of affine schemes in the abstract sense.\n\\begin{thm} \\label{thm:TensLocs}Let $\\mathcal{A}_{V}$ be an affinoid localization of an affinoid $k$-algebra $\\mathcal{A}$. Let $\\mathcal{B}$ be an affinoid $k$-algebra. Then the natural morphism  \n\\[\\mathcal{A}_{V}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{B} \\to \\mathcal{A}_{V}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B} \\]\nis an isomorphism in $\\sD^{-}(\\mathcal{A})$.\nIn particular let $\\mathcal{A}_{W_1}$ and $\\mathcal{A}_{W_2}$ be affinoid localizations of an affinoid algebra $\\mathcal{A}$. Then $\\mathcal{A}_{W_1} \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{A}_{W_2} \\cong \\mathcal{A}_{W_1} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{W_2}.$ Therefore taking $W_1=W_2$, any affinoid localization is a homotopy epimorphism.\n\\end{thm}\n\n{\\bf Proof.}\nFor Weirstrass or Laurent localizations this follows immediately from using induction on Lemma \\ref{lem:easyLW} or the fact that it holds for the more general set of rational localizations which we now consider. When $\\mathcal{A}_{W_1}$ and $\\mathcal{A}_{W_2}$ are rational localizations this has been shown already in Lemma \\ref{lem:LCSrHoEpis}. Suppose now that $V=W_1$ is an rational domain and $W=W_2=V_1 \\cup \\cdots \\cup V_{N}$ is an affinoid domain written as a union of rational domains. For $N=1$ the claim is true. The induction step follows from considering the derived tensor product of $\\mathcal{A}_{V}$ over $\\mathcal{A}$ with the short exact sequence \n\\begin{equation}\\label{equation:ind}0\\to \\mathcal{A}_{W} \\to \\mathcal{A}_{V_1 \\cup \\cdots \\cup V_{N-1}} \\times \\mathcal{A}_{V_N} \\to \\mathcal{A}_{V_1 \\cup \\cdots \\cup V_{N-1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_N} \\to 0.\n\\end{equation}\nAssume by induction that the for some $N>2$ the derived tensor product of the rational localization $\\mathcal{A}_V$ and the affinoid algebra corresponding to the union of $N-1$ or fewer rational domains is equivalent to the ordinary tensor product. This implies that the derived tensor product of $\\mathcal{A}_{V}$ with $\\mathcal{A}_{V_1 \\cup \\cdots \\cup V_{N-1}} \\times \\mathcal{A}_{V_N}$ and with $\\mathcal{A}_{V_1 \\cup \\cdots \\cup V_{N-1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_N} =\\mathcal{A}_{(V_1 \\cap V_{N})\\cup \\dots \\cup (V_{N-1} \\cap V_{N})}$ are equivalent to the ordinary tensor products. Hence the same holds with $\\mathcal{A}_{W}$. Finally, one takes $W$ as $W_1$ in (\\ref{equation:ind}) and considers the derived tensor product with $\\mathcal{A}_{W_2}$ and uses a similar induction to do the general case.\n\n\n\\ \\hfill $\\Box$\n\n\n\n\\begin{rem}\\label{rem:recover}This is analogous to Lemma 2.1.4 (1) of \\cite{TVe5} where it is shown that homotopy Zariski open immersions in the category of affine schemes relative to the closed symmetric monoidal category of abelian groups give precisely the ordinary notion of a Zariski open imemrsions.  In that reference first a special case (inverting a single element of the ring) was shown and the general case follows by their descent formalism.\n\\end{rem}\n\\begin{defn}\nFor any topological space $X$, a {\\it quasi-net} is a set $T$ of subsets of $X$ such that any point $x \\in T$ has a neighborhood of the form $\\cup_{i=1}^{n}V_{i}$ with $x \\in V_{i} \\in T$ for $1 \\leq i \\leq n$. A {\\it net} is a quasi-net $T$ such that such that for every $U,V \\in T$ the set $\\{W \\in T | W \\subset U \\cap V\\}$ is a quasi-net of subsets of $U \\cap V$.  A $k$-analytic space is a locally Hausdorff topological space $X$, a net $\\tau_{0}$ on $X$, a functor $\\phi: \\tau_{0} \\to {\\text{\\bfseries\\sf{Afnd}}}_{k},$ and an invertible natural transformation ${\\text{\\bfseries\\sf{Top}}} \\Longrightarrow {\\text{\\bfseries\\sf{Top}}} \\circ \\phi$. \n\\end{defn}\n\\begin{defn} \nFor any $k$-affinoid algebra $\\mathcal{A}$, the topological space $|\\mathcal{M}(\\mathcal{A})|$ is defined to be the set of non-archimedean bounded semivaluations $| \\ |$ on $\\mathcal{A}$ equipped with the weakest topology such that for each $f \\in \\mathcal{A}$, the maps $|\\mathcal{M}(\\mathcal{A})| \\to \\mathbb{R}_{+}$ defined by sending $| \\ |$ to $|f|$ is continuous.\n\\end{defn}\n\n\\begin{defn}A {\\it k-affinoid space} is a locally ringed space of the form $\\mathcal{M}(\\mathcal{A})=(|\\mathcal{M}(\\mathcal{A})|,\\mathcal{O}_{\\mathcal{M}(\\mathcal{A})})$ where $\\mathcal{A}$ is a $k$-affinoid algebra and $\\mathcal{O}_{\\mathcal{M}(\\mathcal{A})}(U)$ is the limit over $\\mathcal{A}_{V}$, where $V \\subset U$ is a finite union of affinoid domains. The category of $k$-affinoid spaces defined to be a full subcategory of the category of locally ringed spaces of the given form. \n\\end{defn}\nThe category of $k$-affinoid spaces is equivalent to the category  ${\\text{\\bfseries\\sf{Afnd}}}_{k}^{op}$. So we treat $\\mathcal{M}$ as a functor giving this equivalence from ${\\text{\\bfseries\\sf{Afnd}}}_{k}^{op}$ to the category of  $k$-affinoid spaces.\n\\begin{defn}\nA {\\it k-analytic space} consists of a triple $(X,\\tau,\\mathcal{A})$ where $X$ is a locally Hausdorff topological space, $\\tau$ is a net on $X$, and for each $V \\in \\tau$, $\\mathcal{A}(V)$ is a k-affinoid algebra along with a homeomorphism $|\\mathcal{M}(\\mathcal{A}(V))| \\cong V$ (functorially assigned to the elements of $\\tau$) such that if $V, V' \\in \\tau$ and $V' \\subset V$ then $V'$ is an affinoid subdomain of $\\mathcal{M}(\\mathcal{A}(V))$ with coordinate ring $\\mathcal{A}(V')= \\mathcal{A}(V)_{V'}$.  In the event that for every $U \\in \\tau_{2},$ $\\tau_{1}$ restricted to $g^{-1}(U)$ is an atlas of $g^{-1}(U),$ a morphism  $(X_{1},\\tau_{1},\\mathcal{A}_{1}) \\to (X_{2},\\tau_{2},\\mathcal{A}_{2})$ consists of a continuous and $G$-continuous map $g:X_{1} \\to X_{2}$ along with bounded homomorphisms $g^{\\#}_{U,V}:\\mathcal{A}_{2}(U) \\to \\mathcal{A}_{1}(V)$ for every $U \\in \\tau_{2}, V  \\in \\tau_{1}$ with $g(V) \\subset U$ such that for every $V, V' \\in \\tau_{1}$ with $V'\\subset V$ and $U, U' \\in \\tau_{2}$ with $U' \\subset U$ such that $g(V) \\subset U$ and $g(V') \\subset U'$ the diagram \n\\begin{equation}\n\\xymatrix{ \\mathcal{A}_{2}(U) \\ar[d] \\ar[r] & \\mathcal{A}_{1}(V) \\ar[d] \\\\  \\mathcal{A}_{2}(U') \\ar[r] &  \\mathcal{A}_{1}(V') }\n\\end{equation}\ncommutes. We use the terms $k$-analytic space and Berkovich analytic spaces interchangeably. Let ${\\text{\\bfseries\\sf{An}}}_{k}$ denote the category of $k$-analytic spaces.\n\\end{defn}\n\n\n\\begin{defn}\\label{quasinet}A quasi-net on a topological space $X$ is a collection $T$ of subsets of $X$ such that for every $x \\in X$ there is a subset $T_x \\subset T$ with $|T_x| < \\infty$ such that $x \\in \\cap_{V \\in T_x}V$ and there is an open set $U \\subset X$ such that $x \\in U \\subset \\cup_{V \\in T_x} V.$\n\\end{defn}\n\\begin{lem}\\label{easyquasinet}Any finite set $T=\\{V_1,V_2, \\dots V_m \\}$ of closed subsets of a topological space $X$ which cover $X$ is a quasi-net.\n\\end{lem}\n{\\bf Proof.} Given $x \\in X,$ consider the subset $T_x \\subset T$ defined by those subsets in $T$ which contain $x$.  Then $x \\in \\cap_{V \\in T_x}V$. Let $U=X-\\cup_{V \\in T-T_x}V,$ this satisfies the required property.\n\\ \\hfill $\\Box$\n\nConsider ${\\text{\\bfseries\\sf{Afnd}}}^{op}_{k}\/X$, the category of affinoid $k$-analytic spaces over $X$. This means that the objects are pairs $(\\mathcal{M}(\\mathcal{A}),f)$ where $\\mathcal{M}(\\mathcal{A})$ is an affinoid $k$-analytic space and $f:\\mathcal{M}(\\mathcal{A}) \\to X$ is a morphism of $k$-analytic spaces.  The morphisms from $(\\mathcal{M}(\\mathcal{A}_1),f_1)$ to $(\\mathcal{M}(\\mathcal{A}_2),f_2)$ are morphisms to $X$ commuting with the $f_i$.    \n\\begin{lem}\\label{EverythingColimit} Any $k$-analytic space $X$ is a colimit of the category ${\\text{\\bfseries\\sf{Afnd}}}^{op}_{k}\/X$ when considered as a subcategory of ${\\text{\\bfseries\\sf{An}}}_{k}$.\n\\end{lem}\n{\\bf Proof.}\nConsider the family $\\hat{\\tau}$ of all affinoid domains in $|X|$. It is a net and $|X|$ has a maximal $k$-affinoid atlas $\\hat{\\mathcal{A}}$. For each $V \\in \\hat{\\tau}$, $\\hat{\\mathcal{A}}$ assigns $\\mathcal{M}(\\mathcal{A}_{V})\\to X.$  In particular, $\\hat{\\mathcal{A}}$ assigns homeomorphisms $|\\mathcal{M}(\\mathcal{A}_{V})| \\cong V\\subset|X|$ and such that these homeomorphisms satisfy obvious compatibilities. For any other $k$-analytic space $X'$ we have by Exercise 3.2.2 of \\cite{Ber2009} an isomorphism\n\\begin{equation}\n\\Hom(X,X') \\rightarrow \\eq[\\prod_{V \\in  \\hat{\\tau}} \\Hom(\\mathcal{M}(\\mathcal{A}_{V}), X')\\rightrightarrows \\prod_{(V,W) \\in \\hat{\\tau}^{2}}  \\Hom(\\mathcal{M}(\\mathcal{A}_{V}\\widehat{\\otimes_{k}}\\mathcal{A}_{W}), X')\n].\\end{equation}\nTogether with the factorization of this isomorphism as\n\\footnotesize\n\\[\\Hom(X,X') \\to \\lim_{\\mathcal{M} \\in {\\text{\\bfseries\\sf{Afnd}}}^{op}_{k}\/X}\\Hom(\\mathcal{M},X') \\hookrightarrow  \\eq[\\prod_{V \\in  \\hat{\\tau}} \\Hom(\\mathcal{M}(\\mathcal{A}_{V}), X')\\rightrightarrows \\prod_{(V,W) \\in \\hat{\\tau}^{2}}  \\Hom(\\mathcal{M}(\\mathcal{A}_{V}\\widehat{\\otimes_{k}}\\mathcal{A}_{W}),X')]\n\\]\n\\normalsize\nthis implies that the natural morphism \n\\[\\Hom(X,X') \\to \\lim_{\\mathcal{M} \\in {\\text{\\bfseries\\sf{Afnd}}}^{op}_{k}\/X}\\Hom(\\mathcal{M},X')\n\\]\nis an isomorphism. Therefore, $X=\\colim_{\\mathcal{M} \\in {\\text{\\bfseries\\sf{Afnd}}}^{op}_{k}\/X} \\mathcal{M}.$\n\\ \\hfill $\\Box$\n\\subsection{From Banach algebraic geometry to Berkovich geometry}\\label{BSSOCBA}\nConsider the category $\\text{\\bfseries\\sf{C}}={\\text{\\bfseries\\sf{Ban}}}_{k}$ for some valuation field $k$. We have shown in Section \\ref{BanachSpaces} that it is a closed symmetric monoidal quasi-abelian categories with $\\underline{\\Hom}=\\underline{\\Hom}_{k}$ and $\\overline{\\otimes} =\\widehat{\\otimes}_{k}$ with all finite limits and colimits and enough projectives so that we can do algebraic geometry relative to ${\\text{\\bfseries\\sf{Ban}}}_{k}$. In particular the categories of affine schemes over it has certain distinguished morphisms and topologies and we have notions of (Archimedean\/non-Archimedean) Banach schemes Banach (infinity) stacks and $n$-algebraic Banach stacks over an (Archimedean\/non-Archimedean) valuation field $k$.  \nIn the Archimedean case we could compare this geometry to the geometry of complex varieties covered by Stein compact subsets. However, we focus here on the non-Archimedean case. In this section $k$ will be a non-Archimedean valuation field.\n\n\\begin{defn}\\label{def:NDiskrelA}\nLet $\\mathcal{A}$ be a $k$-affinoid algebra. Given $r_{1}, \\cdots, r_{n} \\in \\mathbb{R}$, we can define an $\\mathcal{A}$ algebra \n\\begin{equation}\\mathcal{A}\\{r_1^{-1}T_1, \\dots, r_n^{-1}T_n\\}\n\\end{equation}\nas the completion of $\\mathcal{A}[T_1, \\dots, T_n]$ with respect to the norm \n\\[\\|\\sum a_{I} T^{I}\\|_{r}=\\text{max}_{I}\\{\\|a_{I}\\|_{\\mathcal{A}}r^{I}\\}.\n\\] \n\\end{defn}\nLet $r$ be a real number greater than zero, denote by $\\mathcal{A}_{r}$ the $\\mathcal{A}$ module with norm $\\|a\\|= r\\|a\\|_{\\mathcal{A}}$.\n\\begin{lem}\\label{lem:DiskAlg}$\\mathcal{A}\\{r_1^{-1}T_1, \\dots, r_n^{-1}T_n\\}$ is the symmetric algebra (see subsection \\ref{Symmetric}) on $\\mathcal{V}=\\mathcal{A}_{r_1} \\oplus \\cdots \\oplus \\mathcal{A}_{r_n}$ in ${\\text{\\bfseries\\sf{Mod}}}^{\\leq 1}(\\mathcal{A})$. It can also be seen as the filtered colimit in ${\\text{\\bfseries\\sf{Mod}}}^{\\leq 1}(\\mathcal{A})$ of \n\\[\\sS^{0}(\\mathcal{V}) \\hookrightarrow \\sS^{0}(\\mathcal{V}) \\oplus \\sS^{1}(\\mathcal{V})  \\hookrightarrow \\sS^{0}(\\mathcal{V}) \\oplus \\sS^{1}(\\mathcal{V}) \\oplus \\sS^{2 }(\\mathcal{V})\\hookrightarrow \\cdots \n\\]\nwhere \n\\[\\sS^{m}(\\mathcal{V}) = \\{ \\sum_{|I|=m} a_{I} T^{I} | a_{I} \\in \\mathcal{A}\\}\\]\nequipped with the norm \n\\[\\|\\sum_{|I|=m} a_{I} T^{I}  \\|= \\max_{|I|=m} \\|a_I\\| r^{I}.\n\\]\n\\end{lem}\n{\\bf Proof.} Left to the reader.\n\\ \\hfill $\\Box$\n\n\n\n\n\\begin{lem}\\label{lem:FinTypePres} \n\nA morphism $p: \\mathcal{A} \\to \\mathcal{B}$ in  ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k})$ induces a presentation \n\\begin{equation}\\label{eqn:NiceForm} \\mathcal{B} \\cong \\mathcal{A}\\{\\frac{T_1}{r_1}, \\dots, \\frac{T_g}{r_g}\\}\/(P_1, \\dots, P_r) \n\\end{equation}\nwhere $P_{i} \\in \\mathcal{A}\\{\\frac{T_1}{r_1}, \\dots, \\frac{T_g}{r_g}\\}$\nif and only if $p$ is of finite presentation in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k})$ as defined in Definition \\ref{defn:FinitePres}.\n\\end{lem}\n\n\\ \\hfill $\\Box$\n\nThe following two lemmas are technical results that will be used only in the proof of Theorem \\ref{thm:localForm}.\n\\begin{lem}\\label{SplitProdStr2}Let $\\mathcal{A}, \\mathcal{C}$ be $k$-affinoid algebras considered as objects in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$ and let $f:\\mathcal{A} \\to \\mathcal{C}$ be morphism in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$ which is a strict epimorphism when considered in ${\\text{\\bfseries\\sf{Ban}}}_{k}$ such that there is a $k$-affinoid algebra $\\mathcal{B}$ and the post-composition of $f$ with some homotopy epimorphism $g:\\mathcal{C} \\to \\mathcal{B}$ in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$ is a homotopy epimorphism $h:\\mathcal{A}\\to \\mathcal{B}$ in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$. Then there is a $k$-affinoid algebra $\\mathcal{A}'$ and a isomorphism $\\mathcal{A} \\cong \\mathcal{C} \\times \\mathcal{A}'$ such that the projection to $\\mathcal{C}$ corresponds to $f$ under this isomorphism.\\end{lem}\n{\\bf Proof.} \nThe morphisms \n\\[\\mathcal{A}\\to \\mathcal{C} \\to \\mathcal{B}\n\\]\ninduce morphisms on the derived categories \n\\[\\sD^{-}(\\mathcal{B}) \\to \\sD^{-}(\\mathcal{C}) \\to \\sD^{-}(\\mathcal{A}).\n\\]\nSince the composition is fully faithful and the first morphism is as well, the second morphism must be fully faithful and so by Lemma \\ref{lem:HomotopyMon} we have $\\mathcal{C} \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{C} \\cong \\mathcal{C}$. Let $I=\\ker (f)$. There is a strict, short exact sequence \n\\begin{equation}\\label{eqn:idealAC} 0 \\to I \\to \\mathcal{A} \\to \\mathcal{C} \\to 0.\n\\end{equation}\nIf we consider the derived completed tensor product of (\\ref{eqn:idealAC}) over $\\mathcal{A}$ with $\\mathcal{C}$ we find an exact triangle \n\\[I \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{C} \\to \\mathcal{C} \\to \\mathcal{C} \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{C}\n\\]\nand because the second morphism is an isomorphism, we see that \n$I \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{C}$ is isomorphic to $0$. If we now consider the derived completed tensor product over $\\mathcal{A}$ of (\\ref{eqn:idealAC}) with $I$ we an exact triangle \n\\[I \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} I \\to I \\to I \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{C}\n\\]\nand so we get an isomorphism $I \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}I \\to I$.\nSo we have $I=\\text{image}[I \\widehat{\\otimes}_{\\mathcal{A}}I \\to I]$ and in fact this implies that $I= I^2:= \\text{image}[I \\otimes_{\\mathcal{A}}I \\to I]$. Therefore, there exists an element $e \\in \\mathcal{A}$ such \nthat $e^{2}=e$ and $e\\mathcal{A} =I$. This gives the structure of a $k$-affinoid algebra \nto $I$, which we denote \nby $\\mathcal{A}'=\\mathcal{A}\/(1-e)\\mathcal{A}$. Now because $f$ is \na strict epimorphism, there is a strict short exact sequence \n\\[0 \\to I \\to \\mathcal{A} \\stackrel{f}\\to \\mathcal{C} \\to 0\n\\]\nwhich in fact is split by the morphism of algebras $e:\\mathcal{A} \\to \\mathcal{A}'.$ Therefore, \\[(e,f):\\mathcal{A} \\to \\mathcal{A}' \\times \\mathcal{C}\\] is \nan isomorphism.\n\\ \\hfill $\\Box$\n\\begin{lem}\\label{CoverOfAff}Let $\\mathcal{A}, \\mathcal{B}$ be $k$-affinoid algebras and let $f:\\mathcal{A} \\to \\mathcal{B}$ be a morphism in the category of $k$-affinoid algebras with the property that $|\\mathcal{M}(\\mathcal{A})|$ has a finite covering by affinoid domains $V_{j}$ corresponding to affinoid domain embeddings $\\mathcal{M}(\\mathcal{A}_{V_j})\\to \\mathcal{M}(\\mathcal{A}).$ Suppose also that morphisms $\\mathcal{M}(\\mathcal{A}_{V_j}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B}) \\to \\mathcal{M}(\\mathcal{A}_{V_j})$ are affinoid domain embeddings. Then the morphism $\\mathcal{M}(\\mathcal{B}) \\to \\mathcal{M}(\\mathcal{A})$ is an affinoid domain embedding.\n\\end{lem}\n{\\bf Proof.}\nLet us denote by $U$ the image of $|\\mathcal{M}(\\mathcal{B})|$ inside $|\\mathcal{M}(\\mathcal{A})|.$ We have $\\overline{U} \\cap V_{j} = \\overline{U \\cap V_j} $ for all $j$. Since $U \\cap V_j$ is closed in $V_j$ and $V_j$ is closed in $|\\mathcal{M}(\\mathcal{A})|$ we see that  $U \\cap V_j$ is closed in $|\\mathcal{M}(\\mathcal{A})|$.  Therefore $\\overline{U} \\cap V_{j} = U \\cap V_{j} $ for all $j$. Hence $\\overline{U}=U$ and so $U$ is closed inside $|\\mathcal{M}(\\mathcal{A})|.$\nLet $\\mathcal{A} \\to \\mathcal{C}$ be a bounded homomorphism of affinoid $k$-algebras such that the image of $|\\mathcal{M}(\\mathcal{C})|$ lies in $U$.  We wish to show that the morphism $\\mathcal{M}(\\mathcal{C}) \\to \\mathcal{M}(\\mathcal{A})$ factors through a morphism $\\mathcal{M}(\\mathcal{C}) \\to \\mathcal{M}(\\mathcal{B})$. Notice that $\\mathcal{A}_{V_j} \\to \\mathcal{C} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} $ is a bounded homomorphism of affinoid $k$-algebras such that the image of $\\mathcal{M}(\\mathcal{C} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} )$ lies in $U\\cap V_j$. Therefore, the morphisms $\\mathcal{M}(\\mathcal{C} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} ) \\to \\mathcal{M}(\\mathcal{A}_{V_j} )$ factor in a unique way through morphisms  $\\mathcal{M}(\\mathcal{C} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} )  \\to \\mathcal{M}(\\mathcal{A}_{V_j}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B}).$ When thought of as morphisms   $\\mathcal{M}(\\mathcal{C} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} )  \\to \\mathcal{M}(\\mathcal{B})$ they agree when pulled back to $\\mathcal{M}(\\mathcal{C} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j \\cap V_k} ). $ The preimages of $V_j$ in $|\\mathcal{M}(\\mathcal{C})|$ are analytic domains by \\cite{Te} Exercise 3.2.2 (v). These preimages are the pullback of a quasi-net and therefore form a quasi-net by Lemma \\ref{easyquasinet} and therefore by Exercise 3.2.2 (v) of \\cite{Ber2009} we have a unique morphism $\\mathcal{M}(\\mathcal{C})  \\to \\mathcal{M}(\\mathcal{B})$ which restricts to the morphisms $\\mathcal{M}(\\mathcal{C} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} )  \\to \\mathcal{M}(\\mathcal{B}).$ \nIndeed, this follows from the commutative diagram of exact sequences \n\\footnotesize\n\\begin{equation}\n\\xymatrix{\n0 \\ar[r] & \\Hom(\\mathcal{M}(\\mathcal{C}),\\mathcal{M}(\\mathcal{B})) \\ar[d] \\ar[r]  & \\prod_{j}\\Hom(\\mathcal{M}(\\mathcal{C}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} ),\\mathcal{M}(\\mathcal{B})) \\ar[r] \\ar[d] & \n\\prod_{j,k}\\Hom(\\mathcal{M}(\\mathcal{C}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j\\cap V_k} ),\\mathcal{M}(\\mathcal{B})) \\ar[d] \\\\ \n0 \\ar[r] &\\Hom(\\mathcal{M}(\\mathcal{C}),\\mathcal{M}(\\mathcal{A})) \\ar[r] & \\prod_{j}\\Hom(\\mathcal{M}(\\mathcal{C}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j} ),\\mathcal{M}(\\mathcal{A})) \\ar[r] &  \\prod_{j,k}\\Hom(\\mathcal{M}(\\mathcal{C}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_j\\cap V_k} ),\\mathcal{M}(\\mathcal{A})). }\n\\end{equation}\n\\normalsize\nThis clearly provides the required factorisation. \n\\ \\hfill $\\Box$\n\nIf also, the $\\mathcal{M}(\\mathcal{A}_{V_j})$ are rational in $\\mathcal{M}(\\mathcal{A})$ and  $\\mathcal{M}(\\mathcal{A}_{V_j}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B})$ is rational in $\\mathcal{M}(\\mathcal{A}_{V_j})$ notice that $\\mathcal{M}(\\mathcal{B})$ is a union of the rational domains $\\mathcal{M}(\\mathcal{A}_{V_j}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{B})$ in $\\mathcal{M}(\\mathcal{A}).$\n \n\\begin{lem}\\label{lem:NonExpFP} Let $\\mathcal{A}$ be a $k$-affinoid algebra and suppose there is a morphism $f: \\mathcal{A} \\to \\mathcal{B}$ of finite presentation in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k}).$ Then $\\mathcal{B}$ is a $k$-affinoid algebra.\n\\end{lem}\n{\\bf Proof.}\nBy combining a presentation for $\\mathcal{B}$ over $\\mathcal{A}$ and a presentation for $\\mathcal{A}$ over $k$ one can write $\\mathcal{B}$ as a finite colimit of objects of finite presentation in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k})$. Therefore, $\\mathcal{B}$ has finite presentation in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}^{\\leq 1}_{k})$.\n\\ \\hfill $\\Box$\n\\begin{thm}\\label{thm:localForm}Let $\\mathcal{A}, \\mathcal{B}$ be $k$-affinoid algebras and let $f:\\mathcal{A} \\to \\mathcal{B}$ be a morphism in the category of $k$-affinoid algebras. Assume that $f$ is a homotopy epimorphism (see Definition \\ref{defn:homotopyEpi}) when considered in the category ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$. Then the morphism $\\mathcal{M}(\\mathcal{B}) \\to \\mathcal{M}(\\mathcal{A})$ corresponding to $f$ is an affinoid domain embedding.\\end{thm}\n{\\bf Proof.} \nWe refer here to Temkin's proof \\cite{Te2} of the Gerritzen-Grauert Theorem for morphisms of affinoid algebras.  This theorem, assuming only the epimorphism condition on $f$, produces a finite collection of morphisms of $k$-affinoid algebas $\\mathcal{A} \\to \\mathcal{A}_{V_i}$ corresponding to rational domain embeddings $\\mathcal{M}(\\mathcal{A}_{V_i}) \\to \\mathcal{M}(\\mathcal{A})$ covering $|\\mathcal{M}(A)|$ with the images $V_i.$  The theorem further ensures that the morphisms $\\mathcal{A}_{V_i} \\to \\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i}$ induced from $f$ admit factorzations \\[\\mathcal{A}_{V_i} \\twoheadrightarrow \\mathcal{C}_{i} \\hookrightarrow \\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i}.\\] These factorizations correspond to the composition of the morphism of $k$-affinoid algebras $\\mathcal{C}_{i} \\to (\\mathcal{C}_{i})_{W_i}=\\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i}$ corresponding to Weierstrass domain embeddings  with the surjective morphisms of affinoid $k$-algebras $\\mathcal{A}_{V_i} \\to \\mathcal{C}_{i}$ corresponding to closed immersions. \nTherefore, by Lemma \\ref{lem:LCSrHoEpis} the morphism  $\\mathcal{C}_{i} \\to \\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i}$ is a homotopy epimorphism in the category ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$. Notice that \n\\[(\\mathcal{B}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_i})\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}_{V_i}}(\\mathcal{B}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_i}) \\cong \\mathcal{B}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_i}\n\\]\nbecause homotopy epimorphisms are closed under derived base change. However, applying Lemma \\ref{lem:LCSrHoEpis} we have \n\\[\\mathcal{B}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_i} \\cong \\mathcal{B}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_i}\n\\]\nand so we see that \n\\[(\\mathcal{B}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_i})\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}_{V_i}}(\\mathcal{B}\\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_i}) \\cong \\mathcal{B}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_i}\n\\]\nand so by Lemma \\ref{lem:HomotopyMon} the morphisms $\\mathcal{A}_{V_i} \\to \\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i}$ are homotopy epimorphisms. Also, the morphisms of affinoid algebras corresponding to Weierstrass domain embeddings are injective. Therefore, Lemma \\ref{SplitProdStr2} can applied by choosing the $f$ from that lemma to be the morphism $\\mathcal{A}_{V_i} \\to \\mathcal{C}_{i}$ and $g$ from that lemma to be the morphism $\\mathcal{C}_{i} \\to \\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i}.$ The lemma then tells us that the morphism $\\mathcal{M}(\\mathcal{C}_i) \\to \\mathcal{M}(\\mathcal{A}_{V_i})$ is simply the inclusion of a connected component in a disjoint union of affinoids. Therefore,  $\\mathcal{M}(\\mathcal{C}_i) \\to \\mathcal{M}(\\mathcal{A}_{V_i})$ is an affinoid domain embedding. Because the composition of affinoid domain embeddings is an affinoid domain embedding, we conclude that the morphisms $\\mathcal{M}(\\mathcal{B} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_i})\\to \\mathcal{M}(\\mathcal{A}_{V_i})$ are affinoid domain embeddings as well. By Lemma \\ref{CoverOfAff} we conclude that the original morphism gives an affinoid domain embedding $\\mathcal{M}(\\mathcal{B}) \\to \\mathcal{M}(\\mathcal{A})$.\n\n\\ \\hfill $\\Box$\n\n\n\nFrom now on we write ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A})$ in place of ${\\text{\\bfseries\\sf{Mod}}}_{{\\text{\\bfseries\\sf{Afnd}}}_{k}^{op}}^{RR}(\\mathcal{A})$.\n\\begin{lem}\\label{lem:ConsImpliesSur}\nLet $\\mathcal{A}$ be a $k$-affinoid algebra. Let $\\{f_{i}:\\mathcal{A} \\to \\mathcal{A}_{V_i}\\}_{i \\in I}$ be a family of affinoid localizations such that for some finite set $J \\subset I$ the corresponding family of functors \n\\[{\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}) \\to {\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}_{V_i})\n\\]\nfor $i \\in J$ is conservative. Then the morphism $\\coprod_{i\\in J}\\mathcal{M}(\\mathcal{A}_{V_i}) \\to \\mathcal{M}(\\mathcal{A})$ is surjective.\n\\end{lem}\n{\\bf Proof.}\nWe argue by contradiction.  First assume that $k$ is non-trivially valued and $\\mathcal{A}$ is strictly affinoid. Suppose that the family of functors is conservative and some point $x\\in \\mathcal{M}(\\mathcal{A})$ is not in the image. By Proposition 2.1.15 of \\cite{Ber1990} the subset of points of $y\\in \\mathcal{M}(\\mathcal{A})$ such that $\\ker(|\\;|_{y})$ is a maximal ideal is a dense subset of $\\mathcal{M}(\\mathcal{A})$. Therefore, since the image is the closed set $\\cup_{i\\in J}V_{i}$ we may assume (by changing the point $x$) that $x\\in \\mathcal{M}(\\mathcal{A})$ is not in the image and $\\ker(|\\;|_{x})$ is a maximal ideal. Chose using Proposition 2.2.3 (iii) of \\cite{Ber1990} an affinoid subdomain $W$ of $\\mathcal{M}(\\mathcal{A})$ such that $x \\in W$ and $W \\cap V_{i}$ is empty for all $i \\in J$.  Consider the \n morphism $0 \\to \\mathcal{A}_W$ of ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A})$. It is not an isomorphism but for each $i \\in J$, the pullback to each $\\spec(\\mathcal{A}_{V_i})$ is the isomorphism $0 \\to 0=\\mathcal{A}_{W} \\widehat{\\otimes}_{A} \\mathcal{A}_{V_i}$ of ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}_{V_i})$. This gives a contradiction. For the general case, choose using Proposition 2.1.2 of \\cite{Ber1990}, a valuation field extension $k \\to K$ such that the valuation on $K$ is non-trivial and $\\mathcal{A}\\widehat{\\otimes}_{k}K$ is a strictly $K$-affinoid algebra. Notice that the conservativity assumption on the original family implies by Lemma \\ref{cor:BaseChangeConservHzAR} applied to the base change $\\spec(\\mathcal{A}\\widehat{\\otimes}_{k}K) \\to \\spec(\\mathcal{A})$ that the family of functors $\\{{\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}\\widehat{\\otimes}_{k}K) \\to {\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}_{V_i}\\widehat{\\otimes}_{k}K)\\}_{i \\in J}$ is also conservative. The morphism $\\coprod_{i\\in J}\\mathcal{M}(\\mathcal{A}_{V_i}\\widehat{\\otimes}_{k} K) \\to \\mathcal{M}(\\mathcal{A}\\widehat{\\otimes}_{k}K)$ cannot be surjective because in the commutative diagram, \n\\[\n\\xymatrix{ \\coprod_{i\\in J}\\mathcal{M}(\\mathcal{A}_{V_i}\\widehat{\\otimes}_{k}K) \\ar[d] \\ar[r] & \\coprod_{i\\in J}\\mathcal{M}(\\mathcal{A}_{V_i}) \\ar[d] \\\\  \\mathcal{M}(\\mathcal{A}\\widehat{\\otimes}_{k}K) \\ar[r] &  \\mathcal{M}(\\mathcal{A}) }\\]\nthe horizonal arrows are surjective.  Therefore, we have reduced to the previous case and so the proof is complete.\n\\ \\hfill $\\Box$\n\n\\begin{lem}\\label{lem:AlternatingTate}Consider a (surjective) cover of $X= \\mathcal{M}(\\mathcal{A})$ by a finite collection of affinoid domains $V_i= \\mathcal{M}(A_{V_{i}})$. Then the complex \n\\[\n0 \\to \\mathcal{A} \\to \\prod_{i_1} \\mathcal{A}_{V_{i_1}} \\to \\prod_{i_1 < i_2} \\mathcal{A}_{V_{i_1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_2}} \\to \\cdots \\to  \\mathcal{A}_{V_{1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{2}}\\widehat{\\otimes}_{\\mathcal{A}} \\cdots \\widehat{\\otimes}_{\\mathcal{A}}  \\mathcal{A}_{V_{n}} \\to 0. \n\\]\nis strictly exact.\n\\end{lem}\n{\\bf Proof.} By Proposition 1, section 8.1 of \\cite{BGR}, the inclusion (which is strict) of alternating cochains inside all cochains is a quasi-isomorphism. On the other hand, the complex of all cochains is strictly exact by Proposition 2.2.5 of \\cite{Ber1990}. The conclusion follows.\n\\ \\hfill $\\Box$\n\\begin{lem}\\label{lem:CoversRconservative}Consider a (surjective) cover of $X= \\mathcal{M}(\\mathcal{A})$ by a finite collection of affinoid domains $V_i= \\mathcal{M}(A_{V_{i}})$. Then the corresponding family of functors ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}) \\to {\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}_{V_i})$ is conservative.\n\\end{lem}\n{\\bf Proof.}\nLet $f: \\mathcal{M} \\to \\mathcal{N}$ in ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A})$ be any morphism such that $f_{i}: \\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_i} \\to \\mathcal{N} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_i} $ are isomorphisms for all $i$. \nThe alternating version of the \\v{C}ech-Amitsur complex (see Definition \\ref{defn:Amitsur}) corresponding to the morphism $\\mathcal{A} \\to \\prod_{i=1}^{n} \\mathcal{A}_{V_i}$ is a strictly exact bounded above complex by \\ref{lem:AlternatingTate} and so defines an element of $D^{-}(\\mathcal{A})$ (in fact the $0$ element!).\n\\[0 \\to \\mathcal{A} \\to \\prod_{i_1} \\mathcal{A}_{V_{i_1}} \\to \\prod_{i_1 < i_2} \\mathcal{A}_{V_{i_1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_2}} \\to \\cdots \\to  \\mathcal{A}_{V_{1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{2}}\\widehat{\\otimes}_{\\mathcal{A}} \\cdots \\widehat{\\otimes}_{\\mathcal{A}}  \\mathcal{A}_{V_{n}} \\to 0. \n\\]\nEach object in this complex is acyclic for the functor $\\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}}(-)$ since $\\mathcal{M}$ (being RR-quasi-coherent) is transversal to localizations of $\\mathcal{A}$. Therefore by Proposition \\ref{prop:ProjFProj}, if we apply the derived functor $\\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}}^{\\mathbb{L}}(-)$ we are left with a strictly exact complex \n\\[0 \\to \\mathcal{M} \\to \\prod_{i_1} \\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_1}} \\to \\prod_{i_1<i_2}\\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_2}} \\to \\cdots \\to \\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_{1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{2}}\\widehat{\\otimes}_{\\mathcal{A}} \\cdots \\widehat{\\otimes}_{\\mathcal{A}}  \\mathcal{A}_{V_{n}} \\to 0.\n\\]\nWe can do the same thing for $\\mathcal{N}$. The $f_{i}$ extend uniquely to morphisms of the complexes resolving $\\mathcal{M}$ and $\\mathcal{N}$. Therefore $f$ is an isomorphism. Conversely, if $f$ is an isomorphism, the $f_i$ obviously are as well.\n\\ \\hfill $\\Box$\n\\begin{rem}\\label{rem:TateAcylicity} In the proof of Lemma \\ref{lem:CoversRconservative} we showed the interesting fact that for any $\\mathcal{M} \\in {\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A})$, and any finite cover $\\coprod_{i} \\mathcal{M}(\\mathcal{A}_{V_i}) \\to \\mathcal{M}(\\mathcal{A})$ the complex  \n\\[0 \\to \\mathcal{M} \\to \\prod_{i_1} \\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_1}} \\to \\prod_{i_1<i_2}\\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_2}} \\to \\cdots \\to \\mathcal{M} \\widehat{\\otimes}_{\\mathcal{A}}\\mathcal{A}_{V_{1}} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{2}}\\widehat{\\otimes}_{\\mathcal{A}} \\cdots \\widehat{\\otimes}_{\\mathcal{A}}  \\mathcal{A}_{V_{n}} \\to 0.\n\\]\nis strictly exact (a version of Tate acyclicity). \n\\end{rem}\n\\begin{example}This example was suggested via a correspondence with V. Berkovich. The analogue of Corollary 4.48 is false for finite modules since they are not preserved by push-forward. On the other hand,  RR-quasi-coherent modules are preserved by push-forward (see Lemma \\ref{lem:RRpreserve}) and do detect surjectivity. If $k$ is a field equipped with the trivial valuation and $\\mathcal{A}= k\\{\\frac{x}{r}\\}$ for $0<r<1$ and $\\mathcal{A}_{V}= k\\{\\frac{x}{r'}\\}$ for $0<r'<r$ then $\\mathcal{M}(\\mathcal{A}_{V})\\to \\mathcal{M}(\\mathcal{A})$ is not surjective while the natural morphism $\\mathcal{A} \\to \\mathcal{A}_{V}$ induces an equivalence ${\\text{\\bfseries\\sf{Mod}}}^{f}(\\mathcal{A})\\to {\\text{\\bfseries\\sf{Mod}}}^{f}(\\mathcal{A}_{V})$, as both categories are equivalent to the category of finite modules in the algebraic sense over $k[[x]]$.  On the other hand, it is easy to find objects of ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A})$ which go to zero in ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}_{V})$. For instance, choose a valued extension $K$ of $k$ such that $K$ is non-trivialy valued and $\\mathcal{A}\\widehat{\\otimes}_{k}K$ is strictly affinoid. Chose a point $x \\in \\mathcal{M}(\\mathcal{A}\\widehat{\\otimes}_{k}K) - \\mathcal{M}(\\mathcal{A}_{V}\\widehat{\\otimes}_{k}K)$ such that $\\ker(|\\ |_{x})$ is a closed maximal ideal of $\\mathcal{A}\\widehat{\\otimes}_{k}K$. Choose using Proposition 2.2.3 (iii) of \\cite{Ber1990} an affinoid subdomain $W$ of $\\mathcal{M}(\\mathcal{A}\\widehat{\\otimes}_{k}K)$ such that $W$ does not intersect $\\mathcal{M}(\\mathcal{A}_{V}\\widehat{\\otimes}_{k}K)$ and $x \\in W$. Then we have the non-zero object $(\\mathcal{A}\\widehat{\\otimes}_{k}K)_{W}$ of ${\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A})$. It is the push-forward of $(\\mathcal{A}\\widehat{\\otimes}_{k}K)_{W} \\in {\\text{\\bfseries\\sf{Mod}}}^{RR}(\\mathcal{A}\\widehat{\\otimes}_{k}K)$ along $\\spec(\\mathcal{A}\\widehat{\\otimes}_{k}K) \\to \\spec(\\mathcal{A})$ and therefore RR-quasicoherent by Lemma \\ref{lem:RRpreserve}.  It goes to zero after applying the functor $(-)\\widehat{\\otimes}_{\\mathcal{A}}A_{V}$ because using (\\ref{eqn:basechangeequation}) we have \\[(\\mathcal{A}\\widehat{\\otimes}_{k}K)_{W}\\widehat{\\otimes}_{\\mathcal{A}}A_{V} \\cong (\\mathcal{A}\\widehat{\\otimes}_{k}K)_{W} \\widehat{\\otimes}_{\\mathcal{A}\\widehat{\\otimes}_{k}K}(\\mathcal{A}_{V}\\widehat{\\otimes}_{k}K)\\cong 0.\n\\]\n\\end{example}\n\\subsection{Topologies in the Banach algebraic geometry setting}\n\n\n\n\\begin{thm}\\label{thm:covers}Consider the Berkovich space $\\mathcal{M}(\\mathcal{A})$ for $\\mathcal{A}\\in {\\text{\\bfseries\\sf{Afnd}}}_{k}$. The covers in the weak $G$-topology on $\\mathcal{M}(\\mathcal{A})$, as defined on page 30 of \\cite{Ber1990}, coincide precisely with the covers of $\\spec(\\mathcal{A})$ by affinoids in the formal homotopy Zariski topology on the homotopy transversal subcategory ${\\text{\\bfseries\\sf{Afnd}}}_{k}^{op} \\subset{\\text{\\bfseries\\sf{Aff}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$ (using the terminology of Definition \\ref{defn:fhTVZ} and Proposition  \\ref{hfTV1}).\n\\end{thm}\n{\\bf Proof.}\nSay that we are given a cover of $\\mathcal{M}(\\mathcal{A})$ by rational domains which has a finite subcover.  Lemma \\ref{lem:LCSrHoEpis} tells us that the affinoid domains correspond to homotopy epimorphisms $\\mathcal{A} \\to \\mathcal{B}_{i}$ for some $i \\in I$ in ${\\text{\\bfseries\\sf{Afnd}}}^{op}_{k}$. Lemma \\ref{lem:CoversRconservative} tells us that the family of functors indexed by the finite subset $J \\subset I$ corresponding to the finite subcover is conservative. In the other direction suppose we are given a cover of $X$ in the formal homotopy Zariski topology. It has a finite conservative sub-cover which must be surjective by Lemma \\ref{lem:ConsImpliesSur}. Every element of it must be a subset of $\\mathcal{M}(\\mathcal{A})$  because every morphism in the cover is a monomorphism. In fact, Theorem \\ref{thm:localForm} implies that every morphism in the cover is an affinoid domain embedding. This concludes the proof.\n\\ \\hfill $\\Box$\n\n\n\\begin{lem}\\label{lem:IsSheaf}\nLet $\\mathcal{A}, \\mathcal{B}_{i}$ be $k$-affinoid algebras and let $\\{f_{i}:\\mathcal{A} \\to \\mathcal{B}_{i}\\}_{i\\in I}$ be a family of morphisms of $k$-affinoid algebras which is a formal homotopy Zariski open cover in the category ${\\text{\\bfseries\\sf{Afnd}}}_{k}^{op}$. Chose a finite subset $J \\subset I$ with the conservative property. Then for any $k$-analytic space $X$ we have \n\\begin{equation}\\label{equation:gluingdiagram}\\Hom(\\mathcal{M}(\\mathcal{A}),X)=\\eq [\\prod_{i \\in J}\\Hom(\\mathcal{M}(\\mathcal{B}_{i}),X) \\rightrightarrows \\prod_{i \\in J,j \\in J}\\Hom(\\mathcal{M}(\\mathcal{B}_{i} \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{B}_{j}), X)].\n\\end{equation}\nTherefore, the presheaf $\\straighth_{X}$ is a sheaf in the \nformal homotopy Zariski topology on the homotopy transversal subcategory ${\\text{\\bfseries\\sf{Afnd}}}_{k}^{op} \\subset{\\text{\\bfseries\\sf{Aff}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$.\n\\end{lem}\n{\\bf Proof.}\nLemma \\ref{lem:ConsImpliesSur} and Theorem \\ref{thm:localForm} imply that the family of morphisms indexed by $J$ corresponds to a finite cover by affinoid domains. Lemma \\ref{easyquasinet} implies that it is a quasinet. Exercise 3.2.2 (v) of \\cite{Ber2009} now tells us that Equation \\ref{equation:gluingdiagram} is valid.\n\n\\ \\hfill $\\Box$\n\\noindent\n\nUnlike the rest of this article, the following theorem is only significant given the results in \\cite{BeKr2}.\n\\begin{thm}\\label{thm:BigRestriction} The topology of Definition \\ref{defn:TVtopologyIntro} restricted to the category ${\\text{\\bfseries\\sf{Afnd}}}_{k}^{op}$ agrees with the weak G-topology. \n\\end{thm}\n\n{\\bf Proof.}\n\nIn light of Theorems \\ref{thm:TensLocs} and \\ref{thm:localForm} we only need to understand why condition (2) of Definition \\ref{defn:TVtopologyIntro} is equivalent to a finite collection of affinoid subdomains $V_i$ covering every point of an affinoid $\\mathcal{M}(\\mathcal{A})$. Suppose first that the union of the $V_i$ is all of $\\mathcal{M}(\\mathcal{A})$. Say we have a morphism $\\mathcal{M} \\to \\mathcal{N}$ in $D^{-}(\\mathcal{A})$ and we know that the induced morphism $\\mathcal{M}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_i}\\to \\mathcal{N}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_i}$ is an isomorphism in  $D^{-}(\\mathcal{A})$ for all $i$. For any $i_1 < i_2 < \\cdots < i_p$ we get an isomorphism $\\mathcal{M}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_{i_1, \\dots, i_p}}\\to \\mathcal{N}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}\\mathcal{A}_{V_{i_1, \\dots, i_p}}$ where \\[\\mathcal{A}_{V_{i_1, \\dots, i_p}}= \\mathcal{A}_{V_{i_1}} \\widehat{\\otimes}_{\\mathcal{A}} \\cdots  \\widehat{\\otimes}_{\\mathcal{A}} \\mathcal{A}_{V_{i_1}}= \\mathcal{A}_{V_{i_1}} \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\cdots  \\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}} \\mathcal{A}_{V_{i_1}}.\\] Therefore, we get an isomorphism from \n\\begin{equation}\\label{eqn:Mres}\\mathcal{M}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}( \\prod_{i_1 } \\mathcal{A}_{V_{i_1}} \\to \\prod_{i_1 < i_2} \\mathcal{A}_{V_{i_1, i_2}}  \\to \\cdots \\to  \\mathcal{A}_{V_{1,2,...,n}}  \\to 0) \n\\end{equation}\nto\n\\begin{equation}\\label{eqn:Nres}\\mathcal{N}\\widehat{\\otimes}^{\\mathbb{L}}_{\\mathcal{A}}( \\prod_{i_1 } \\mathcal{A}_{V_{i_1}} \\to \\prod_{i_1 < i_2} \\mathcal{A}_{V_{i_1, i_2}}  \\to \\cdots \\to  \\mathcal{A}_{V_{1,2,...,n}}  \\to 0) \n\\end{equation}\nin $D^{-}(\\mathcal{A})$. Since by Lemma \\ref{lem:AlternatingTate} the natural morphism from $\\mathcal{A}$ \nto \n\\[\n\\prod_{i_1 } \\mathcal{A}_{V_{i_1}} \\to \\prod_{i_1 < i_2} \\mathcal{A}_{V_{i_1, i_2}}  \\to \\cdots \\to  \\mathcal{A}_{V_{1,2,...,n}}  \\to 0\n\\]\nis an isomorphism in $D^{-}(\\mathcal{A})$,  (\\ref{eqn:Mres}) and (\\ref{eqn:Nres}) are equivalent to $\\mathcal{M}$ and $\\mathcal{N}$ respectively. Therefore, we can conclude that the original morphism $\\mathcal{M} \\to \\mathcal{N}$ is an isomorphism. Conversely, suppose that there is a point $x\\in \\mathcal{M}(\\mathcal{A})$ which is not in any of the $V_{i}$. Suppose first that $k$ is non-trivially valued and that $\\mathcal{A}$ is strictly $k$-affinoid. Chose as in Lemma \\ref{lem:ConsImpliesSur} a subdomain $W$ of $\\mathcal{M}(\\mathcal{A})$ such that $x \\in W$ and $W$ does not intersect any of the $V_i$. Then $\\mathcal{A}_{W}\\to 0$ is a morphism in $D^{-}(\\mathcal{A})$ but for each $V_{i}$ the derived pullback gives an isomorphism $\\mathcal{A}_{W} \\widehat{\\otimes}_{A}^{\\mathbb{L}}\\mathcal{A}_{V_{i}} \\cong  \\mathcal{A}_{W} \\widehat{\\otimes}_{A}\\mathcal{A}_{V_{i}} \\cong 0 \\to 0$. For the general case, choose using Proposition 2.1.2 of \\cite{Ber1990}, a valuation field extension $k \\to K$ such that the valuation on $K$ is non-trivial and $\\mathcal{A}\\widehat{\\otimes}_{k}K$ is a strictly $K$-affinoid algebra. Notice that the conservativity assumption on the original family implies by Lemma \\ref{lem:DerivedBaseChangeConserv} applied to the base change $\\spec(\\mathcal{A}\\widehat{\\otimes}_{k}K) \\to \\spec(\\mathcal{A})$ that the family of functors $\\{D^{-}(\\mathcal{A}\\widehat{\\otimes}_{k}K) \\to D^{-}(\\mathcal{A}_{V_i}\\widehat{\\otimes}_{k}K)\\}_{i \\in J}$ is also conservative. The morphism $\\coprod_{i\\in J}\\mathcal{M}(\\mathcal{A}_{V_i}\\widehat{\\otimes}_{k} K) \\to \\mathcal{M}(\\mathcal{A}\\widehat{\\otimes}_{k}K)$ cannot be surjective because in the commutative diagram, \n\\[\n\\xymatrix{ \\coprod_{i\\in J}\\mathcal{M}(\\mathcal{A}_{V_i}\\widehat{\\otimes}_{k}K) \\ar[d] \\ar[r] & \\coprod_{i\\in J}\\mathcal{M}(\\mathcal{A}_{V_i}) \\ar[d] \\\\  \\mathcal{M}(\\mathcal{A}\\widehat{\\otimes}_{k}K) \\ar[r] &  \\mathcal{M}(\\mathcal{A}) }\\]\nthe horizonal arrows are surjective.  Therefore, we have reduced to the previous case and so the proof is complete. \n\\ \\hfill $\\Box$\n\nFor the next lemma, we will need the notion of continuous and cocontinuous functors. These are defined in the appendix in Definition \\ref{defn:ContCocont}. \n\\begin{lem}\\label{lem:inclprops}The inclusion functor \n\n\\[{\\text{\\bfseries\\sf{Afnd}}}^{op}_{k} \\to {\\text{\\bfseries\\sf{Aff}}}({\\text{\\bfseries\\sf{Ban}}}_{k}) \n\\]\nis fully faithful and continuous with respect to the Zariski, formal Zariski, homotopy Zariski, formal homotopy Zariski and fpqc topologies.\n\\end{lem}\n{\\bf Proof.}\nMost of these properties are more or less obvious at this point. The fiber products correspond to completed tensor products of affinoid algebras or commutative algebra objects in ${\\text{\\bfseries\\sf{Ban}}}_{k}$. The fact that these functors take covers to covers in the various topologies follows from the definition of covers on the domain of the functors. The fully faithful property is discussed in \\ref{lem:extension} of the appendix.\n  \n\\hfill $\\Box$\n\nRecall that ${\\text{\\bfseries\\sf{An}}}_{k}$ is the category of Berkovich spaces over a non-Archimedean valuation field $k$.\n\\begin{thm}\\label{thm:embedding} There is a fully faithful embedding from ${\\text{\\bfseries\\sf{An}}}_{k}$ to the categories of presheaves of sets on ${\\text{\\bfseries\\sf{Afnd}}}^{op}_{k}$ and ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})^{op}.$\nThis embedding induces a fully faithful embedding \n\\[{\\text{\\bfseries\\sf{An}}}_{k} \\hookrightarrow  {\\text{\\bfseries\\sf{Sch}}}(({\\text{\\bfseries\\sf{Afnd}}}^{op}_{k})^{fhZar}) \n\\].\nFor any affinoid algebras $\\mathcal{A}$ and $\\mathcal{B}$, the scheme assigned to\n$\\mathcal{M}(\\mathcal{A})$ evaluates on $\\spec(\\mathcal{B})$ to $\\Hom_{{\\text{\\bfseries\\sf{Afnd}}}_{k}}(\\mathcal{A},\\mathcal{B})$.\n\\end{thm}\n{\\bf Proof.}\nLet $X$ be a $k$-analytic space. Consider the contravariant functors \n\\[\\straighth_{X}:{\\text{\\bfseries\\sf{Afnd}}}^{op}_{k} \\to {\\text{\\bfseries\\sf{Set}}}\n\\]\ndefined by \n\\[\\straighth_{X}(\\spec(\\mathcal{A}))=\\Hom_{{\\text{\\bfseries\\sf{An}}}_{k}}(\\mathcal{M}(\\mathcal{A}),X)\n\\]\nfor $\\mathcal{A} \\in {\\text{\\bfseries\\sf{Afnd}}}_{k}$.\nThey form a functor\n\\begin{equation}\\label{equation:AnToPre}{\\text{\\bfseries\\sf{An}}}_{k} \\stackrel{\\straighth}\\to {\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Afnd}}}^{op}_{k})\n\\end{equation}\nwhich can be seen the Yoneda embedding for ${\\text{\\bfseries\\sf{An}}}_{k}$ followed by the restriction functor ${\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{An}}}_{k}) \\to {\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Afnd}}}^{op}_{k})$. \nWe would like to know that for every $X_{1},X_{2} \\in {\\text{\\bfseries\\sf{An}}}_{k}$ that \n\\[\\Hom_{{\\text{\\bfseries\\sf{An}}}_{k}}(X_{1},X_{2}) \\to \\Hom_{{\\text{\\bfseries\\sf{Pr}}}({\\text{\\bfseries\\sf{Afnd}}}^{op}_{k})}(\\straighth_{X_1},\\straighth_{X_2})\n\\]\nis an isomorphism. The fact that the functor $\\straighth$ is fully faithfull follows immediately from Lemma \\ref{EverythingColimit} which says that $X$ is the final object in the category of morphisms $\\spec(\\mathcal{A}) \\to X$ where $\\mathcal{A} \\in {\\text{\\bfseries\\sf{Afnd}}}_{k}$ and the fact that for any full subcategory ${\\tC} \\subset {\\tD}$ such that every object $d$ in ${\\tD}$ is a colimit of objects in ${\\tC}$ mapping to $d$, the functor $\\straighth$ embeds ${\\tC}$ fully and faithfully into the category of presheaves of sets on ${\\tD}$.  This fact can be found in \\cite{SGA4}, Exp I, Prop 7.2. \nConsider the fully faithful, continuous and cocontinuous functor from Lemma \\ref{lem:inclprops}\n\\[\\su: {\\text{\\bfseries\\sf{Afnd}}}^{op}_{k} \\to {\\text{\\bfseries\\sf{Aff}}}({\\text{\\bfseries\\sf{Ban}}}_{k}). \n\\]\n\nwhich is left adjoint to the restriction functor. \n\nIt is clear by the definition of Berkovich analytic spaces that we actually get a fully faithful embedding \n\\[{\\text{\\bfseries\\sf{An}}}_{k} \\hookrightarrow  {\\text{\\bfseries\\sf{Sch}}}(({\\text{\\bfseries\\sf{Afnd}}}^{op}_{k})^{fhZar}) \n\\]\nand when we restrict the scheme to the subcategory ${\\text{\\bfseries\\sf{Afnd}}}_{k}^{op}$ this assignment agrees with the standard functor of points in the category ${\\text{\\bfseries\\sf{An}}}_{k}$.\n\\ \\hfill $\\Box$\n\n\\begin{rem}\nNote that exactly the same argument shows that the category of rigid analytic spaces (which contains $k$-analytic spaces as a full subcategory) embeds fully faithfully into ${\\text{\\bfseries\\sf{Sch}}}(({\\text{\\bfseries\\sf{Afnd}}}^{op}_{k})^{fhZar}).$ \n\\end{rem}\n\n\\begin{rem}\nIn order to embed $k$-analytic spaces and rigid analytic spaces into a catgeory of Banach schemes, ${\\text{\\bfseries\\sf{Sch}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$, we need to work in the derived setting as the homotopy Zariski topology is only well defined on simplicial Banach algebras. As ${\\text{\\bfseries\\sf{Afnd}}}_{k}$ is homotopy Zariski transversal this topology restricts to the non-derived site of affinoids. The derived approach is pursued in \\cite{BeKr2}.  \n\\end{rem}\n\n\\subsection{Huber Points}\n\nLet $X$ be an object of ${\\text{\\bfseries\\sf{Sch}}}(({\\text{\\bfseries\\sf{Afnd}}}_{k})^{op})^{hZar})$. Define a category ${\\text{\\bfseries\\sf{hZar}}}(X)$ as the full sub-category of ${\\text{\\bfseries\\sf{Sh}}}(({\\text{\\bfseries\\sf{Afnd}}}_{k})^{op})^{hZar})\/X$ whose objects are $u:Y\\to X$ with $u$ a homotopy Zariski open immersion. This category \nis a locale and inherits a topology: a family $\\{Y_i\\to Y\\}$ is a covering if and only if $\\coprod Y_i\\to Y$ is an epimorphism of sheaves. We have the full subcategory ${\\text{\\bfseries\\sf{hZarAff}}}(X)$ whose objects are $u:Y\\to X$ where $Y$ is affine. \nThe continuous inclusion ${\\text{\\bfseries\\sf{hZarAff}}}(X)\\to {\\text{\\bfseries\\sf{hZar}}}(X)$ induces an equivalence of categories of sheaves:\n${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{hZar}}}(X))\\cong {\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{hZarAff}}}(X))$. \n\nThe fact that ${\\text{\\bfseries\\sf{hZar}}}(X)$ is a locale and that the topology on it \nis generated by a quasi-compact  pre-topology (as covering families in ${\\text{\\bfseries\\sf{Sh}}}({\\text{\\bfseries\\sf{hZarAff}}}(X))$ are finite), implies that \n${\\text{\\bfseries\\sf{hZar}}}(X)$ is equivalent as a locale to the locale of open subsets of a topological space $|X|$. Using this we can view $X$ a ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k})$-valued ringed space:\n$(|X|,\\mathcal{O}_X)$, where $\\mathcal{O}_X$ is a sheaf valued in ${\\text{\\bfseries\\sf{Comm}}}({\\text{\\bfseries\\sf{Ban}}}_{k}).$\n\n\n\\begin{lem}\nLet $X$ be a Berkovich space. ${\\text{\\bfseries\\sf{hZarAff}}}(X)$ is the rigid analytic site of $X$.\n\\end{lem}\n\n{\\bf Proof.} This is a rephrasing of Theorem \\ref{thm:covers}.\n\\hfill $\\Box$\n\n\\begin{thm}\nLet $X$ be a Berkovich space. The space $|X|$ is the space of Huber points.\n\\end{thm}\n{\\bf Proof.} From the previous lemma we know that the locale\ncorresponding to $X$ is the rigid analytic one. From Huber \\cite{Hu}, we see that the topological space corresponding to this locale is the space of Huber points of $X$.\n\\hfill $\\Box$\n\n  \n\n\\section{Work in progress}\nIn future work, we intend to embed rigid analytic spaces and Huber spaces into the categories of schemes over the opposite category to Banach algebras. We also intend develop a variant which will work with complete, convex bornological algebras, instead of Banach algebras or Fr\\'echet algebras or other types of topological vector spaces. Similarly to Paugam \\cite{Pa} we would like to handle the Archimedean and non-Archimedean cases with a single language. Also with the same goal, Bambozzi has defined and studied affinoid bornological algebras and dagger algebras in his thesis \\cite{Bam}. The categories based on convex bornological spaces are nicer than those based on ${\\text{\\bfseries\\sf{Ban}}}_{k}$ in that they have arbitrary limits and colimits. The bornological categories contain the categories based on Banach spaces or Fr\\'echet spaces via fully faithful embeddings. A lot of work in this direction of bornological geometry and representation theory has been done by Meyer \\cite{M} in the Archimedean case over $\\mathbb{C}$ and by Houzel and others \\cite{H,H2} in the case of a valuation field where the valuation is not the trivial valuation. The extension to the case of a trivially valued field is more difficult, especially in the definition of completeness. Therefore, we could have done our constructions for convex bornological vector spaces over an arbitrary field or for complete convex bornological vector spaces over a non-discretely valued field.  In another direction we think that categories of topological spaces, topological manifolds, differentiable manifolds and real analytic manifolds could also be described in terms of this style of algebraic geometry. This could fit in well with the algebraic analysis of Kashiwara, Schapira, Saito and others. Non-commutative versions may be possible as well along the lines of \\cite{So,KR}. It would also be nice to be able to work over rings instead of fields, for instance the rings of $p$-adic intgers $\\mathbb{Z}_{p}$ and the integers equipped with either the discrete or the standard norm and do types of analytic geometry over these rings. We are developing a formalism of derived analytic stacks along the lines of following To\\\"{e}n and Vezzosi's approach \\cite{TVe} or Lurie's approach \\cite{L1,L2}. It will work simultaneously in the Archimedean and non-Archimedean setting. In particular we need to define a cotangent complex and the notion of smooth and \\'{e}tale morphisms and compare these notions with the notions defined by Berkovich.\nWe also believe that a non-Archimedean versions of both Lurie's Tanakian duality theorem \\cite{L3} and (after developing differential graded categories of modules) the algebrization theorem \\cite{TV2} of To\\\"{e}n and Vaqui\\'{e} should be provable in our framework. We plan to prove a version of the Beilinson-Bernstein localization theorem over certain non-Archimedean fields in our context which would be a generalisation of the work of \\cite{AW} over $\\mathbb{Q}_{p}$. Finally, we believe that when working over a finite field with trivial valuation, our formalism will be useful for studying $G$-bundles over an algebraic surface or Kac-Moody bundles over an algebraic curve with an eye towards representation theory as in \\cite{BrKa}, \\cite{K}, \\cite{GP}. This is based on the relationship between $G$-bundles on a punctured formal neighborhood of a curve in surface and twisted $G((t))$-bundles on the curve. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe classical Monge's transport problem refers to the problem of moving one distribution of mass onto another as efficiently as possible, where the efficiency criterion is expressed in terms of the average distance transported. It originates from a paper by G. Monge, \\textit{M\\'emoire sur la th\\'eorie des d\\'eblais et des remblais}, in 1781. Rephrased and generalized in modern mathematical terms, we are given two Borel probability measures $\\mu$ and $\\nu$ on a metric space $(X,d)$ and we want to minimize\n\\begin{equation*}\n T \\mapsto \\int_X d(x,T(x))\\,d\\mu(x)\n\\end{equation*}\namong all transport maps $T$ from $\\mu$ to $\\nu$, i.e., all $\\mu$-measurable maps $T:X\\to X$ such that $T_\\#\\mu=\\nu$, meaning that $\\nu(B)=\\mu(T^{-1}(B))$ for all Borel set $B$. \n\nIn this paper, we are interested in Monge's transport problem in the Heisenberg group $(\\mathbb{H}^n,d)$ equipped with its Carnot-Carath\\'eodory distance. We prove the existence of an optimal transport map between two compactly supported Borel probability measures $\\mu$ and $\\nu$ on $\\mathbb{H}^n$ assuming that the first measure $\\mu$ is absolutely continuous with respect to the Haar measure $\\mathcal{L}^{2n+1}$ of $\\mathbb{H}^n$. \n\n\\begin{thm} \\label{mainthm}\nLet $\\mu$ and $\\nu$ be two compactly supported Borel probability measures on $\\mathbb{H}^n$. Assume that $\\mu \\ll \\mathcal{L}^{2n+1}$. Then there exists an optimal transport map solution to Monge's transport problem between $\\mu$ and $\\nu$, i.e., a $\\mu$-measurable map $T:\\mathbb{H}^n\\rightarrow\\mathbb{H}^n$ such that $T_\\#\\mu=\\nu$ and\n\\begin{equation*}\n\\int_{\\mathbb{H}^n} d(x,T(x))\\,d\\mu(x) = \\inf_{S_\\#\\mu=\\nu} \\int_{\\mathbb{H}^n} d(x,S(x))\\,d\\mu(x).\n\\end{equation*}\n\\end{thm}\n\nMonge's transport problem in $\\mathbb{R}^n$ equipped with a distance induced by a norm has already been widely investigated. A first attempt to solve this problem goes back to the work of Sudakov \\cite{sudakov}. It was however discovered some years later that the proof in \\cite{sudakov} was not completely correct. In \\cite{evans-gangbo} a PDE-based alternative to Sudakov's approach has been developed. The authors prove the existence of an optimal transport map in $\\mathbb{R}^n$ equipped with the Euclidean norm under the assumptions that $\\operatorname{spt} \\mu \\cap \\operatorname{spt} \\nu = \\emptyset$, $\\mu$, $\\nu \\ll \\mathcal{L}^{n}$ with Lipschitz densities with compact support. Existence results for general absolutely continuous measures $\\mu$, $\\nu$ with compact support have been obtained independently in \\cite{cfm} and \\cite{TrudWang} and have been extended to a Riemannian setting in \\cite{FeldMcC}. The existence of a solution to Monge's transport problem assuming only that the initial measure $\\mu$ is absolutely continuous has been proved in \\cite{ambrosio}, see also \\cite{ap}, \\cite{akp}, \\cite{caravenna1}. All these later proofs roughly involve a Sudakov-type dimension reduction argument, via different technical implementations though, and require some regularity assumptions about the norm $\\mathbb{R}^n$ is endowed with. For some time, it seemed that there were indeed some borderline cases about the norms that could not be attacked through these techniques. \n\nRecently another approach that does not go through Sudakov-type arguments and in particular does not require disintegration of measures has been developed in \\cite{champion-dePascale}, see also \\cite{champion-dePascale-first}, to solve Monge's transport problem for general norms in $\\mathbb{R}^n$. This approach relies on rather simple but powerful density results. In the present paper we follow closely this approach. We basically show that a very similar strategy can be implemented in the context of the Heisenberg group equipped with its Carnot-Carath\\'eodory distance. The main features that play a role in this approach are that $(\\mathbb{H}^n, d, \\mathcal{L}^{2n+1})$ is a doubling polish metric measure space, a non-branching geodesic space and satisfies a so-called Measure Contraction Property. It is very likely that this approach can be extended to more general metric measure spaces, see Section~\\ref{extensions}. We have chosen however to present the particular case of the Heisenberg group for simplicity, this space being moreover an instructive explicit example of non Riemannian space.~\\footnote{The preprint~\\cite{bianc-cav} which appeared during the completion of the present paper also addresses Monge's transport problem in metric spaces with a geodesic distance using a Sudakov-type dimension reduction argument and disintegration of measures.} \n\nThe strategy starts by considering the nowadays classical relaxation of Monge's transport problem proposed by Kantorovich. In Kantorovich' formulation one considers transport plans, i.e., Borel probability measures on $X \\times X$ with first and second marginals $\\mu$ and $\\nu$ respectively. Denoting by $\\Pi(\\mu,\\nu)$ the class of all transport plans, one wants to minimize\n\\begin{equation*}\n \\gamma \\mapsto \\int_{X\\times X} d(x,y)\\,d\\gamma(x,y)\n\\end{equation*}\namong all $\\gamma \\in \\Pi(\\mu,\\nu)$. Due to the linearity of the constraint $\\gamma \\in \\Pi(\\mu,\\nu)$, weak topologies provide existence of optimal transport plans. As a classical fact it turns out that whenever an optimal transport plan is induced by a $\\mu$-measurable map $T$, i.e., can be written in the form $(I \\otimes T)_\\sharp\\mu$ where $(I \\otimes T)(x) := (x,T(x))$, then $T$ is an optimal transport map between $\\mu$ and $\\nu$ solution to Monge's transport problem. We follow here this scheme seeking after optimal transport plans that will be shown to be eventually induced by $\\mu$-measurable maps.\n\nIn our present context we first prove that any optimal transport plan is concentrated on a set of pairs of points that are connected by a unique minimal curve and that these transport rays cannot bifurcate (see Section~\\ref{sect:optplanning}). Next, following ideas already introduced in the literature and more specifically here inspired by~\\cite{sant}, one introduces variational approximations (see Section~\\ref{varapprox}). This procedure allows to select optimal transport plans with specific properties. These transport plans will eventually be proved to be induced by $\\mu$-measurable maps. This procedure is here essentially twofold. On one hand it allows to select optimal transport plans that are solution to a secondary variational problem. This secondary variational problem prescribes the geometry of transport rays. The selected transport plans are indeed shown to be monotonic along transport rays (see Lemma~\\ref{pi2}). On the other hand, given a transport plan, one can in our context interpolate between its first and second marginal in a natural way (see Subsection~\\ref{sect-interpolation}). Absolute continuity and more importantly $L^\\infty$-estimates on the density of the interpolations will play an important role and one can indeed prove $L^\\infty$-estimates on the interpolations in the approximating variational problems (see Proposition~\\ref{e:Linftydensityestimates}). These estimates rely on the so-called Measure Contraction  Property of $\\mathbb{H}^n$. In the limit one will  eventually get suitable $L^\\infty$-estimates on the interpolations constructed from optimal transport plans selected through the variational approximation procedure. Next we note that some properties of plans with absolutely continuous first marginal proved in~\\cite{champion-dePascale} can be easily generalized to our setting (see Section~\\ref{lebpoints}). These properties are independent of the transport problem. They rely on the notion of Lebesgue points of functions and Lebesgue points of sets, notions which make sense for instance in any doubling metric measure space. Together with the above mentioned $L^\\infty$-estimates on the interpolations, one can in particular prove density estimate on the transport set of selected optimal transport plans in the same way as in~\\cite{champion-dePascale} (see Section~\\ref{main}). All together, namely combining this later density estimate on the transport set (Lemma~\\ref{mainlemma}) with Lemma~\\ref{dens2} and remembering the monotonicity along transport rays (Lemma~\\ref{pi2}) it turns out that the selected transport plans are necessarily induced by a transport map as eventually proved in Theorem~\\ref{mainthmbis}.\n\nThe paper is organized as follows. In Section~\\ref{opttrans} we recall classical facts about optimal transportation for later use. In Section~\\ref{prelimhn} we describe the Heisenberg group focusing on the features that will be needed in this paper. In Section~\\ref{sect:optplanning} we prove geometric properties of optimal transport plans and the monotonicity along transport rays of solutions to  the secondary variational problem. The variational approximations are introduced and studied in Section~\\ref{varapprox}. In Section~\\ref{lebpoints} we state in our framework properties of plans with absolutely continuous first marginal proved in~\\cite{champion-dePascale} in $\\mathbb{R}^n$. This section, independent of the transport problem, contains density results that play an essential role in the strategy followed here. In Section~\\ref{main} we prove lower bounds on the density, in some suitable sense, of the transport set of optimal transport plans selected through the variational approximations. We conclude in Section~\\ref{conclusion} proving that the selected transport plans are induced by a transport map. We discuss in the final Section~\\ref{extensions} some possible extensions of this approach to other spaces.\n\n\n\\section{Preliminaries on optimal transportation} \\label{opttrans}\n\nWe recall some well-known facts about optimal transportation confining ourselves to statements that will fit our needs in the rest of the paper. More general versions of these results hold in more general contexts. We refer to e.g. \\cite{villani} and the references therein.\n\nLet $(X,d)$ be a Polish space, i.e., a complete and separable metric space. We denote by $\\mathcal P(X)$ the set of all Borel probability measures on $X$ and by $\\mathcal P_c(X)$ the set of Borel probability measures on $X$ with compact support. The weak topology we consider on $\\mathcal P(X)$ is the topology induced by convergence against bounded and continuous test functions (or narrow topology).\n\n\\subsection{Kantorovich transport problem} \nLet $\\mu$, $\\nu\\in\\mathcal P(X)$. We denote by \n\\begin{equation*}\n \\Pi(\\mu,\\nu) := \\{ \\gamma \\in \\mathcal P(X \\times X);\\, (\\pi_1)_\\sharp\\gamma = \\mu,\\, (\\pi_2)_\\sharp\\gamma = \\nu \\}\n\\end{equation*}\nthe set of all transport plans between $\\mu$ and $\\nu$. Here $\\pi_1$, $\\pi_2: X\\times X \\rightarrow X$ denote the canonical projections on the first and second factor respectively.\n\nGiven $c:X\\times X \\rightarrow [0,+\\infty]$ a lower semicontinuous cost function, we look at Kantorovich transport problem between $\\mu$ and $\\nu$ with cost $c$:\n\\begin{equation} \\label{e:MK}\n \\min_{\\gamma \\in \\Pi(\\mu,\\nu)} \\int_{X\\times X} c(x,y)\\,d\\gamma(x,y).\n\\end{equation}\nAs a classical fact, existence of solutions to \\eqref{e:MK} follows from the weak compactness of $\\Pi(\\mu,\\nu)$ together with the lower semicontinuity of the functional to be minimized. We call them optimal transport plans.\n\n\\subsection*{Cyclical monotonicity} We say that a set $\\Gamma\\subset X\\times X$ is $c$-cyclically monotone if \n\\begin{equation*}\n\\sum_{i=1}^N c(x_i,y_i) \\leq \\sum_{i=1}^N c(x_{i+1},y_i)\n\\end{equation*}\nwhenever $N\\geq 2$ and $(x_1,y_1), \\dots, (x_N,y_N)\\in\\Gamma$.\n\n\\begin{thm} \\label{ccycl}\nLet $\\gamma \\in \\Pi(\\mu,\\nu) $ be an optimal transport plan and assume that $\\int_{X\\times Y} c(x,y)\\,d\\gamma <+\\infty$. Then $\\gamma$ is concentrated on a $c$-cyclically monotone Borel set.\n\\end{thm}\n\n\\subsection*{Dual formulation - Kantorovich potentials}\nLet $\\psi:X\\rightarrow \\mathbb{R} \\cup \\{-\\infty\\}$. We say that $\\psi$ is $c$-concave if $\\psi\\not\\equiv -\\infty$ and if there exists $\\varphi:X \\rightarrow \\mathbb{R} \\cup \\{-\\infty\\}$, $\\varphi\\not\\equiv -\\infty$, such that \n\\begin{equation*}\n\\psi(x) = \\inf_{y\\in X} c(x,y) - \\varphi(y).\n\\end{equation*}\n\n\\begin{thm}  \\label{duality}\nIn addition to the previous assumptions, assume that $c$ is real-valued and that \n\\begin{equation*}\n \\forall\\, (x,y)\\in X\\times X, \\qquad c(x,y) \\leq a(x) + b(y)\n\\end{equation*}\nfor some $a\\in L^1(\\mu)$ and $b\\in L^1(\\nu)$. Then one has\n\\begin{equation} \\label{e:duality}\n \\min_{\\gamma\\in \\Pi(\\mu,\\nu)} \\int_{X\\times X} c(x,y)\\,d\\gamma(x,y) \n= \\max \\int_{X} \\psi(x)\\,d\\mu(x) + \\int_{X} \\psi^c(y)\\,d\\nu(y)\n\\end{equation}\nwhere the above maximum is taken among all $c$-concave functions $\\psi$ and $ \\psi^c(y) := \\inf_{x\\in X} c(x,y) - \\psi(x)$. \n\\end{thm}\n\n\\begin{defi} [Kantorovich potentials]\n We say that $\\psi:X\\rightarrow \\mathbb{R} \\cup \\{-\\infty\\}$ is a Kantorovich potential if $\\psi$ is a $c$-concave maximizer for the right-hand side of \\eqref{e:duality}.\n\\end{defi}\n\n\\begin{thm} \\label{caracterisation_opt_planning}\nWith the same assumptions as in Theorem~\\ref{duality}, let $\\psi$ be a Kantorovich potential. Then $\\gamma\\in \\Pi(\\mu,\\nu)$ is an optimal transport plan if and only if \n\\begin{equation*}\n c(x,y) = \\psi(x) + \\psi^c(y) \\qquad \\gamma-\\text{a.e. in } X\\times X.\n\\end{equation*}\n\\end{thm}\n\nWe will use these results for various cost functions. In the particular case which is the core of this paper and where $c(x,y) = d(x,y)$ and $\\mu$, $\\nu \\in \\mathcal P_c(X)$, one can rephrase these results in terms of 1-Lipschitz Kantorovich potentials. More precisely, set \n\\begin{equation*}\n \\text{Lip}_1(d) := \\{u:X\\rightarrow \\mathbb{R};\\quad |u(x) - u(y)| \\leq d(x,y) \\quad \\forall\\, x, y \\in X\\}.\n\\end{equation*}\n\n\\begin{thm} \\label{1lip_potential}\nLet $\\mu$, $\\nu \\in \\mathcal P_c(X)$. Then one can find a Kantorovich potential $u\\in \\text{Lip}_1(d) $ so that\n\\begin{equation*}\n \\min_{\\gamma\\in \\Pi(\\mu,\\nu)} \\int_{X\\times X} d(x,y)\\,d\\gamma(x,y) \n= \\int_{X} u(x)\\,d\\mu(x) - \\int_{X} u(y)\\,d\\nu(y)\n\\end{equation*}\nand \n$\\gamma\\in \\Pi(\\mu,\\nu)$ is an optimal transport plan solution to Kantorovich transport problem~\\eqref{e:MK} between $\\mu$ and $\\nu$ with cost $c(x,y) = d(x,y)$ if and only if \n\\begin{equation*}\nu(x) - u(y) = d(x,y)  \\qquad \\gamma-\\text{a.e. in } X\\times X.\n\\end{equation*}\n\\end{thm}\n\n\\subsection{Transport problem} Let $\\mu$, $\\nu\\in\\mathcal P(X)$. We say that a $\\mu$-measurable map $T:X\\rightarrow X$ is a transport map between $\\mu$ and $\\nu$ if $T_\\sharp\\mu=\\nu$, i.e., $\\nu(B)=\\mu(T^{-1}(B))$ for\nall Borel set $B$. \n\nGiven $c:X\\times X \\rightarrow [0,+\\infty[$ a continuous cost function, we look at the transport problem between $\\mu$ and $\\nu$ with cost $c$:\n\\begin{equation}  \\label{e:M}\n \\min_{T_\\#\\mu=\\nu} \\int_{X} c(x,T(x))\\,d\\mu(x).\n\\end{equation}\n\nWe say that a transport plan $\\gamma\\in \\Pi(\\mu,\\nu)$ is induced by a transport if there exists a $\\mu$-measurable map $T:X\\rightarrow X$ such that $(I \\otimes T)_\\sharp\\mu = \\gamma$ where $(I \\otimes T)(x) := (x,T(x))$. Such a map is automatically a transport map between $\\mu$ and $\\nu$. We also recall that if a transport plan $\\gamma$ is concentrated on a $\\gamma$-measurable graph then $\\gamma$ is induced by a transport.\n\n\\begin{thm} [Optimal transport plans versus optimal transport maps] \\hfill \\label{plan-transport}\n\n(i) Assume that $\\gamma$ is an optimal transport plan solution to Kantorovich transport problem~\\eqref{e:MK} and that $\\gamma$ is induced by transport $T$. Then $T$ is an optimal transport map solution to the transport problem~\\eqref{e:M}.\n\n(ii) Assume that any optimal transport plan solution to Kantorovich transport problem~\\eqref{e:MK} is induced by transport. Then there exists a unique optimal transport map solution to the transport problem~\\eqref{e:M}.\n \\end{thm}\n\n\n\\section{Preliminaries on $\\mathbb{H}^n$} \\label{prelimhn}\n\nWe consider the Heisenberg group $\\mathbb{H}^n$ equipped with its Carnot-Carath\\'eodory distance. Endowed with this distance $\\mathbb{H}^n$ is a polish geodesic and non-branching metric space and a doubling metric measure space when equipped with its Haar measure.\n\n\\subsection{The Heisenberg group}\nThe Heisenberg group $\\mathbb{H}^n$ is a connected, simply connected Lie group with stratified Lie algebra. We identify it with $\\mathbb{C}^n \\times \\mathbb{R}$ equipped with the group law \n\\begin{equation*}\n[\\zeta,t] \\cdot [\\zeta',t'] := [\\zeta + \\zeta', t+t'+ 2 \\sum_{j=1}^n \\operatorname{Im} \\zeta_j \\overline\\zeta'_j]\n\\end{equation*}\nwhere $\\zeta = (\\zeta_1,\\dots,\\zeta_n)$, $\\zeta' = (\\zeta'_1,\\dots,\\zeta'_n)\\in\\mathbb{C}^n$ and $t$, $t'\\in \\mathbb{R}$. The unit element is 0 and the center of the group is \n\\begin{equation*}\nL := \\{[0,t] \\in \\mathbb{H}^n ;\\; t \\in \\mathbb{R}\\}.\n\\end{equation*}\nThere is a natural family of dilations $\\delta_r$ on $\\mathbb{H}^n$ defined by $\\delta_r([\\zeta,t]):= [r\\zeta,r^2 t]$. These dilations are group homomorphisms. \n\nWe may also identify $\\mathbb{H}^n$ with $\\mathbb{R}^{2n+1}$ via the correspondence $[\\zeta,t]= (\\xi,\\eta,t)$ where $\\xi=(\\xi_1,\\dots,\\xi_n)$, \n$\\eta=(\\eta_1,\\dots,\\eta_n) \\in \\mathbb{R}^n$, $t\\in \\mathbb{R}$ and $\\zeta = (\\zeta_1,\\dots,\\zeta_n)\\in\\mathbb{C}^n$ with $\\zeta_j = \\xi_j + i \\eta_j$. The horizontal subbundle of the tangent bundle is defined by\n\\begin{equation*}\n\\mathcal{H} := \\operatorname{span} \\left\\{X_j; \\, j=1,\\dots,n\\right\\} \\oplus \\operatorname{span} \\left\\{Y_j; \\, j=1,\\dots,n\\right\\}\n\\end{equation*}\nwhere the left invariant vector fields $X_j$ and $Y_j$ are given by\n\\begin{equation*}\nX_j := \\partial_{\\xi_j} + 2 \\eta_j \\partial_t \\,, \\quad Y_j := \\partial_{\\eta_j} - 2 \\xi_j \\partial_t.\n\\end{equation*}\nVector fields in $\\mathcal{H}$ will be called horizontal vector fields. Setting $T:=\\partial_t$, the only non trivial bracket relations are $[X_j,Y_j] = -4 T$ hence $\\operatorname{span} \\{T\\} = [ \\mathcal{H},\\mathcal{H}]$ and the Lie algebra $\\mathcal{H}^n$ of $\\mathbb{H}^n$ admits the stratification $\\mathcal{H}^n = \\mathcal{H} \\oplus \\operatorname{span} \\{T\\}$.\n\nThe Lebesgue measure $\\mathcal{L}^{2n+1}$ on $\\mathbb{H}^n \\thickapprox \\mathbb{R}^{2n+1}$ is a Haar measure of the group. It is $(2n+2)$-homogeneous with respect to the dilations, \n\\begin{equation*}\n \\mathcal{L}^{2n+1}(\\delta_r(A)) = r^{2n+2} \\mathcal{L}^{2n+1}(A)\n\\end{equation*}\nfor all Borel set $A$ and all $r>0$.\n\n\\subsection{Carnot-Carath\\'eodory distance} The Carnot-Carath\\'eodory distance on $\\mathbb{H}^n$ is defined by \n\\begin{equation} \\label{e:cc-dist}\n d(x,y) = \\inf \\{ length_{g_0} (\\gamma); \\; \\gamma \\text{ horizontal } C^1 \\text{-smooth curve joining } x \\text{ to } y\\},\n\\end{equation}\nwhere a $C^1$-smooth curve is said to be horizontal if, at every point, its tangent vector belongs to the horizontal subbundle of the tangent bundle and $g_0$ is the left invariant Riemannian metric which makes $(X_1,\\dots,X_n,Y_1,\\dots,Y_n,T)$ an orthonormal basis. For a general presentation of Carnot-Carath\\'eodory spaces, see e.g. \\cite{bel}, \\cite{mont}. \n\nThe topology induced by this distance is the original (Euclidean) topology on $\\mathbb{H}^n \\thickapprox (\\mathbb{R}^{2n+1},g_0)$ and $(\\mathbb{H}^n,d)$ is a  complete metric space. The distance is left invariant and 1-homogeneous with respect to the dilations,\n\\begin{equation*}\nd(x \\cdot y, x\\cdot z) = d(y,z) \\quad \\text{and} \\quad d(\\delta_r(y), \\delta_r(z)) = r\\,d(y,z)\n\\end{equation*}\nfor all $x$, $y$, $z\\in\\mathbb{H}^n$ and all $r>0$. It follows in particular that $B(x,r) = x\\cdot \\delta_r (B(0,1))$ and hence \n\\begin{equation} \\label{e:measball}\n \\mathcal{L}^{2n+1}(B(x,r)) = c_n \\, r^{2n+2}\n\\end{equation}\nfor all $x\\in \\mathbb{H}^n$, all $r>0$ and where $c_n := \\mathcal{L}^{2n+1}(B(0,1))>0$. The measure $\\mathcal{L}^{2n+1}$ is in particular a doubling measure on $(\\mathbb{H}^n,d)$. For more details about doubling metric measure spaces, see e.g. \\cite{heinonen}.\n\nEndowed with its Carnot-Carath\\'eodory distance $\\mathbb{H}^n$ is a geodesic space, i.e., for all $x$, $y\\in\\mathbb{H}^n$, there exists a curve $\\sigma \\in C([a,b],\\mathbb{H}^n)$ such that $\\sigma(a)=x$, $\\sigma(b)=y$ and $d(x,y) = l(\\sigma)$ where \n\\begin{equation*}\n l(\\sigma) = \\sup_{N\\in  \\mathbb{N}^*} \\sup_{a=t_0\\leq \\dots \\leq t_N = b} \\sum_{i=0}^{N-1} d(\\sigma(t_i),\\sigma(t_{i+1})).\n\\end{equation*}\nUp to a reparameterization one can always assume that length minimizing curves $ \\sigma$ are parameterized proportionally to arc-length, i.e., \n\\begin{equation*}\nd(\\sigma(s),\\sigma(s')) = v \\,(s'-s)\n\\end{equation*}\nfor all $s<s' \\in [a,b]$, where $v:=d(\\sigma(a),\\sigma(b))\/(b-a)$ is the (constant) speed of the curve. As a convention we will use throughout this paper the terminology \\textit{minimal curves} to denote length minimizing curves parameterized proportionally to arc-length.\n\n\\begin{defi}[Minimal curves] \nWe say that a continuous curve $\\sigma:[a,b]\\rightarrow\\mathbb{H}^n$ is a minimal curve if $l(\\sigma) = d(\\sigma(a),\\sigma(b))$ and $\\sigma$ is parameterized proportionally to arc-length.\n \\end{defi}\n\nIn general Carnot-Carath\\'eodory spaces, issues about uniqueness and regularity of minimal curves between any two points as well as issues about the regularity of the distance function to a given point could be delicate. In the specific case of the Heisenberg group, equations of all minimal curves can be explicitly computed and exploited to overcome these difficulties. We recall below the description of minimal curves in $\\mathbb{H}^n$, see e.g. \\cite{gaveau}, \\cite{ar}. We set\n\\begin{equation} \\label{e:omega}\n \\Omega := \\{(x,y)\\in \\mathbb{H}^n\\times \\mathbb{H}^n;\\,\\, x^{-1}\\cdot y \\not \\in L\\}.\n\\end{equation}\n\n\\begin{thm} [Minimal curves in $\\mathbb{H}^n$] \\label{geod} \nMinimal curves in $\\mathbb{H}^n$ are horizontal $C^1$-smooth curves such that the infimum in \\eqref{e:cc-dist} is achieved. One has the more precise description:\n\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\begin{enumerate}\n\n\\item Non trivial minimal curves starting from 0 and parameterized on $[0,1]$ are all curves $\\sigma_{\\chi,\\varphi}$ for some $\\chi \\in \\mathbb{C}^n\\setminus\\{0\\}$ and $\\varphi\\in [-2\\pi,2\\pi]$ where \n\\begin{equation*}\n\\sigma_{\\chi,\\varphi}(s) = [i \\dfrac{(e^{-i\\varphi s} -1) \\chi}{\\varphi}, 2 |\\chi|^2 \\,\\dfrac{\\varphi s - \\sin (\\varphi s)}{\\varphi^2}]\n \\end{equation*}\nif $\\varphi\\in [-2\\pi,2\\pi]\\backslash \\{0\\}$ and \n\\begin{equation*}\n \\sigma_{\\chi,\\varphi}(s) = [\\chi s, 0]\n\\end{equation*}\nif $\\varphi =0$. Moreover one has $|\\chi| = d(0,\\sigma_{\\chi,\\varphi}(1))$.\n\n\\item For all $(x,y)\\in \\Omega$, there is a unique minimal curve $x\\cdot \\sigma_{\\chi,\\varphi}$ between $x$ and $y$ for some $\\chi \\in \\mathbb{C}^n\\setminus\\{0\\}$ and some $\\varphi\\in\\, (-2\\pi,2\\pi)$ and one has $|\\chi| = d(x,y)$.\n\n\\item \nIf $(x,y)\\not\\in \\Omega$, $x^{-1}\\cdot y = [0,t]$ for some $t\\in \\mathbb{R}^*$, there are infinitely many minimal curves between $x$ and $y$. These curves are all curves of the form $x \\cdot \\sigma_{\\chi,2\\pi}$ if $t>0$, $x \\cdot \\sigma_{\\chi,-2\\pi}$ if $t<0$, for all $\\chi \\in \\mathbb{C}^n$ such that $|\\chi| = \\sqrt{\\pi |t|}$.\n\\end{enumerate}\n\\end{thm}\n\nHere and in the following, $|\\chi| = (\\sum_{j=1}^n |\\chi_j|^2)^{1\/2}$ for $\\chi = (\\chi_1,\\dots,\\chi_n)\\in\\mathbb{C}^n$. In particular it follows from this description that $(\\mathbb{H}^n,d)$ is non-branching.\n\n\\begin{prop} [Non-branching property of $\\mathbb{H}^n$] \\label{nonbranching}\n The space $(\\mathbb{H}^n,d)$ is non-branching, i.e., any two minimal curves which coincide on a non trivial interval coincide on the whole intersection of their intervals of definition. \n\\end{prop}\n\nEquivalently for any quadruple of points $z$, $x$, $y$, $y'\\in \\mathbb{H}^n$, if $z$ is a midpoint of $x$ and $y$ as well as a midpoint of $x$ and $y'$, then $y=y'$. \n\nThe next lemma collects some differentiability properties of the distance function to a given point to be used later. For $y\\in \\mathbb{H}^n$, we set $L_y:= y\\cdot L$.\n\n\\begin{lem} \\label{prop-distcc} Let $y\\in \\mathbb{H}^n$ and set $d_y(x) := d(x,y)$. Then the function $d_y$ is of class $C^{\\infty}$ on $\\mathbb{H}^n \\backslash L_y$ (equipped with the usual differential structure when identifying $\\mathbb{H}^n$ with $\\mathbb{R}^{2n+1}$). Moreover one has \n\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\begin{enumerate}\n\n\\item $|\\nabla_H d_y(x)| = 1$ for all $x \\in \\mathbb{H}^n \\backslash L_y$ where $$\\nabla_H d_y(x) := (X_1 d_y(x)+i Y_1 d_y(x), \\dots, X_n d_y(x)+i Y_n d_y(x)).$$ \n\n\\item If $\\nabla d_y(x) =  \\nabla d_{y'}(x)$ and $d(x,y) = d(x,y')$ for some $x \\in \\mathbb{H}^n \\backslash ( L_y\\cup L_{y'})$, then $y=y'$. Here $\\nabla = (\\partial_{\\xi_1},\\cdots,\\partial_{\\xi_n},\\partial_{\\eta_1},\\cdots,\\partial_{\\eta_n}, \\partial_t)$ denotes the classical gradient when identifying $\\mathbb{H}^n$ with $\\mathbb{R}^{2n+1}$. \n \\end{enumerate}\n\\end{lem}\n\n\\begin{proof} Set $\\Phi(\\chi,\\varphi):=\\sigma_{\\chi,\\varphi}(1)$ where $\\sigma_{\\chi,\\varphi}$ is given in Theorem~\\ref{geod}. This map is a $C^\\infty$-diffeomorphism from $\\mathbb{C}^n\\setminus\\{0\\} \\times (-2\\pi,2\\pi)$ onto $\\mathbb{H}^n\\setminus L$ (see e.g. \\cite{monti}, \\cite{ar}, \\cite{juillet}). If $x = \\Phi(\\chi,\\varphi) \\in \\mathbb{H}^n\\setminus L$ with $(\\chi,\\varphi) \\in \\mathbb{C}^n\\setminus\\{0\\} \\times (-2\\pi,2\\pi)$, one has $d_0(x) = |\\chi|$ and \n\\begin{equation*}\n\\nabla_H d_0(x) = \\dfrac{\\chi}{|\\chi|} e^{-i\\varphi} \\quad \\text{and} \\quad \\partial_t d_0(x) = \\dfrac{\\varphi}{4|\\chi|},\n\\end{equation*}\nsee \\cite[Lemma 3.11]{ar}. Next, by left invariance, we have $d_y(x) = d_0(y^{-1}\\cdot x)$, $\\nabla_H d_y(x) = \\nabla_H d_0(y^{-1}\\cdot x)$ and $\\partial_t d_y(x) = \\partial_t d_0(y^{-1}\\cdot x)$ if $x\\in \\mathbb{H}^n \\backslash L_y$ and the lemma follows easily.\n\\end{proof}\n\n\\subsection{Interpolation between measures} \\label{sect-interpolation} The notion of interpolation constructed from a transport plan between any two measures will be one of the key notion to be used later. To define it in our geometrical context, we first fix a measurable selection of minimal curves, i.e., a Borel map $S:\\mathbb{H}^n\\times\\mathbb{H}^n \\rightarrow C([0,1], \\mathbb{H}^n)$ such that for all $x$, $y\\in\\mathbb{H}^n$, $S(x,y)$ is a minimal curve joining $x$ and $y$. The existence of such a measurable recipe to join any two points in $\\mathbb{H}^n$ by a minimal curve follows from general theorems about measurable selections, see e.g. \\cite[Chapter 7]{villani}. Next we set $e_t(\\sigma) := \\sigma(t)$ for all $\\sigma \\in C([0,1],\\mathbb{H}^n)$ and $t\\in [0,1]$. In particular $e_t(S(x,y))$ denotes the point lying at distance $t \\, d(x,y)$ from $x$ on the selected minimal curve $S(x,y)$ between $x$ and $y$. \n\n\\begin{defi} \\label{interpolation} \n Let $\\mu$, $\\nu\\in \\mathcal P(\\mathbb{H}^n)$ and let $\\gamma \\in \\Pi(\\mu,\\nu)$. The interpolations between $\\mu$ and $\\nu$ constructed from $\\gamma$ are defined as the family $((e_t \\circ S)_\\sharp \\gamma))_{t\\in [0,1]}$ of Borel probability measures on $\\mathbb{H}^n$. \n\\end{defi}\n\nNote that these interpolations depend a priori on the measurable selection $S$ of minimal curves. This is actually not a serious issue for our purposes. We will moreover always consider interpolations constructed from transport plans that are concentrated on the set $\\Omega$ on which $S(x,y)$ is nothing but the unique minimal curve between $x$ and $y$. Note also for further reference that $S\\lfloor_\\Omega$ is continuous.\n\n\\subsection{Intrinsic differentiability} Intrinsic differentiability properties of real-valued Lipschitz functions on $\\mathbb{H}^n$, namely a Rademacher's type theorem, will be useful when considering 1-Lipschitz Kantorovich potentials. This theorem is a particular case of a more general result due to P. Pansu. We say that a group homomorphism $g :\\mathbb{H}^n \\rightarrow \\mathbb{R}$ is homogeneous if $g(\\delta_r(x)) = r\\,g(x)$ for \nall $x\\in\\mathbb{H}^n$ and all $r >0$. \n\n\\begin{defi}\n We say that a map $f:\\mathbb{H}^n \\rightarrow \\mathbb{R}$ is Pansu-differentiable at $x \\in \\mathbb{H}^n$ if there exists \nan homogeneous group homomorphism $g :\\mathbb{H}^n \\rightarrow \\mathbb{R}$ such that\n\\begin{equation*}\n\\lim_{y \\rightarrow x} \\frac{f(y) - f(x) - g(x^{-1} \\cdot y)}{d(y,x)} = 0.\n\\end{equation*}\nThe map $g$ is then unique and will be denoted by $D_H f(x)$.\n\\end{defi}\n\nIf $f:\\mathbb{H}^n \\rightarrow \\mathbb{R}$ is Pansu-differentiable at $x \\in \\mathbb{H}^n$ then the maps $s \\mapsto f(x \\cdot \\delta_s[e_j, 0])$, resp. $s \\mapsto f(x\\cdot \\delta_s[e_{n+j}, 0])$, are differentiable at $s=0$ and if we denote the corresponding derivatives by $X_jf(x)$, resp. $Y_jf(x)$, then \n\\begin{equation*}\nD_H f(x)(\\xi,\\eta,t) = \\sum_{j=1}^n \\xi_j X_jf(x) + \\eta_j Y_jf(x).\n\\end{equation*}\nHere $e_j = (\\delta_{1}^{j},\\dots,\\delta_{n}^{j})\\in \\mathbb{C}^n$ and $e_{n+j} = (i\\delta_{1}^{j},\\dots,i\\delta_{n}^{j})\\in \\mathbb{C}^n$. Using similar notations as in the classical smooth case, we then set $\\nabla_H f(x) := (X_1 f(x)+i Y_1 f(x), \\dots, X_n f(x)+i Y_n f(x))$.\n\n\\begin{thm}[Pansu-differentiability theorem] \\label{PansuRademacher} \\cite{pansu}\nLet $f:(\\mathbb{H}^n,d) \\rightarrow \\mathbb{R}$ be a $C$-Lipschitz function. Then, for $\\mathcal{L}^{2n+1}$-a.e. $x\\in\\mathbb{H}^n$, the function $f$ is Pansu-differentiable at $x$ and $|\\nabla_H f(x)|\\leq C$.\n\\end{thm}\n\nThe next lemma will be used to prove that any optimal transport plan is concentrated on the set $\\Omega$.\n\n\\begin{lem} \\label{uniquegeod}\n Let $u\\in \\text{Lip}_1(d)$, $x\\in\\mathbb{H}^n$ be such that $u$ is Pansu-differentiable at $x$ with $|\\nabla_H u(x)| \\leq 1$ and let $y\\in\\mathbb{H}^n$ be such that $u(x) - u(y) = d(x,y)$. Then there exists a unique minimal curve between $x$ and $y$.\n\\end{lem}\n\n\\begin{proof}\n Let $\\sigma:[0,1]\\rightarrow\\mathbb{H}^n$ be a minimal curve between $x$ and $y$. Then $\\sigma$ is a horizontal $C^1$-smooth curve and if $\\sigma(t) = (\\sigma_1(t),\\dots,\\sigma_{2n+1}(t)) \\in \\mathbb{H}^n \\thickapprox \\mathbb{R}^{2n+1}$, one has for all $t\\in [0,1]$,\n\\begin{equation*}\n \\dot\\sigma(t) = \\sum_{j=1}^n \\dot\\sigma_j(t)\\, X_j(\\sigma(t)) + \\dot\\sigma_{n+j}(t)\\, Y_j(\\sigma(t))\n\\end{equation*}\nand $|\\dot\\sigma_H(t)| = d(x,y)$ where $\\dot\\sigma_H(t) := (\\dot\\sigma_1(t) +i\\,\\dot\\sigma_{n+1}(t),\\dots,\\dot\\sigma_n(t) +i\\,\\dot\\sigma_{2n}(t))\\in\\mathbb{C}^n$. On the other hand, one has \n\\begin{equation*}\n u(x) - u(\\sigma(t)) = d(x,\\sigma(t)) = t\\, d(x,y)\n\\end{equation*}\nfor all $t\\in [0,1]$. Differentiating this equality with respect to $t$, we get\n\\begin{equation*}\n \\sum _{j=1}^n \\dot\\sigma_j(0)\\, X_ju(x) + \\dot\\sigma_{n+j}(0) \\, Y_ju(x)= \\dfrac{d}{dt}\\,u(\\sigma(t))_{|_{t=0}} = - d(x,y).\n\\end{equation*}\nAll together, it follows that \n\\begin{equation*}\nd(x,y) = |\\sum _{j=1}^n \\dot\\sigma_j(0)\\, X_ju(x) + \\dot\\sigma_{n+j}(0) \\, Y_ju(x)| \\leq  |\\nabla_H u(x)|\\, |\\dot\\sigma_H(0)| \\leq d(x,y).\n\\end{equation*}\nIn particular, there is equality in all the previous inequalities which implies in turn that $\\dot\\sigma_H(0) = - \\,d(x,y) \\,\\nabla_H u (x)$. On the other hand one knows from Theorem~\\ref{geod} that $\\sigma = x\\cdot \\sigma_{\\chi,\\varphi}$ for some $\\chi \\in \\mathbb{C}^n\\setminus\\{0\\}$ and $\\varphi\\in\\, [-2\\pi,2\\pi]$. In particular one has $\\dot\\sigma_H(0) = \\chi$. It follows that $\\chi = - \\,d(x,y) \\,\\nabla_H u (x)$ is uniquely determined hence there is a unique minimal curve joining $x$ and $y$ according once again to the description given in Theorem~\\ref{geod}.\n\\end{proof}\n\n\n\\section{Properties of $\\Pi_1(\\mu,\\nu)$ and $\\Pi_2(\\mu,\\nu)$} \\label{sect:optplanning}\n\nLet $\\mu$, $\\nu \\in \\mathcal P_c(\\mathbb{H}^n)$ be fixed. We denote by $\\Pi_1(\\mu,\\nu)$ the set of optimal transport plans solution to Kantorovich transport problem~\\eqref{e:MK} between $\\mu$ and $\\nu$ with cost $c(x,y) = d(x,y)$. \n\nWe first prove some geometric properties of optimal transport plans. These properties follow from the behavior of minimal curves in $(\\mathbb{H}^n,d)$. In the next lemma, we prove that any optimal transport plan is concentrated on the set $\\Omega$ (see \\eqref{e:omega}) of pair of points that are connected by a unique minimal curve.\n\n\\begin{lem} \\label{pi1.1}\n Let $\\gamma\\in \\Pi_1(\\mu,\\nu)$ and assume that $\\mu\\ll\\mathcal{L}^{2n+1}$. Then for $\\gamma$-a.e. $(x,y)$, there exists a unique minimal curve between $x$ and $y$.\n\\end{lem}\n\n\\begin{proof}\nLet $u\\in \\text{Lip}_1(d)$ be a Kantorovich potential associated to Kantorovich transport problem~\\eqref{e:MK} between $\\mu$ and $\\nu$ with cost $c(x,y) = d(x,y)$ (see Section~\\ref{opttrans} and Theorem~\\ref{1lip_potential} there). Since $u\\in \\text{Lip}_1(d)$, we know from Theorem \\ref{PansuRademacher} that for $\\mathcal{L}^{2n+1}$-a.e., and  hence $\\mu$-a.e., $x\\in\\mathbb{H}^n$, $u$ is Pansu-differentiable at $x$ with $|\\nabla_H u(x)|\\leq 1$. Then the conclusion follows from Lemma~\\ref{uniquegeod} since $u(x) - u(y) = d(x,y)$ for $\\gamma$-a.e. $(x,y)$ (see Theorem~\\ref{1lip_potential}). \n\\end{proof}\n\nThe next lemma says that minimal curves used by an optimal transport plan cannot bifurcate. It follows essentially from the non-branching property of $(\\mathbb{H}^n,d)$.\n\n\\begin{lem} \\label{pi1.2}\n Let $\\gamma\\in \\Pi_1(\\mu,\\nu)$. Then $\\gamma$ is concentrated on a set $\\Gamma$ such that the following holds. For all $(x,y)\\in \\Gamma$ and $(x',y')\\in \\Gamma$ such that $x\\not=y$ and $x\\not=x'$, if $x'$ lies on a minimal curve between $x$ and $y$ then all points $x$, $x'$, $y$ and $y'$ lie on the same minimal curve. More precisely, there exists a minimal curve $\\sigma:[a,b]\\rightarrow\\mathbb{H}^n$ such that $x=\\sigma(a)$, $y=\\sigma(t)$ for some $t\\in \\,(a,b]$, $x'=\\sigma(s)$ for some $s\\in\\, (a,t]$ and $y'=\\sigma(t')$ for some $t'\\in [s,b]$.\n\\end{lem}\n\n\\begin{proof} Let $(x,y)\\in \\mathbb{H}^n \\times \\mathbb{H}^n$ and $(x',y')\\in \\mathbb{H}^n \\times \\mathbb{H}^n$ such that $x\\not=y$ and $x'\\not=x$. Assume that $x'\\in \\sigma((0,d(x,y)])$ where $\\sigma:[0,d(x,y)] \\rightarrow \\mathbb{H}^n$ is a unit-speed minimal curve between $x$ and $y$. Let $\\sigma'$ be a unit-speed minimal curve between $x'$ and $y'$ parameterized on $[d(x,x'),d(x,x')+d(x',y')]$. Assume moreover that \n\\begin{equation*} \n d(x,y) + d(x',y') \\leq d(x,y')+d(x',y).\n\\end{equation*}\nRecall that this holds true for $\\gamma$-a.e. $(x,y)$ and $(x',y')$ by Theorem \\ref{ccycl}. Then the curve $\\tilde \\sigma :[0, d(x,x')+d(x',y')]\\rightarrow\\mathbb{H}^n$ which coincides with $\\sigma$ on $[0,d(x,x')]$ and $\\sigma'$ on $[d(x,x'),d(x,x')+d(x',y')]$ is a length minimizing curve between $x$ and $y'$. Indeed, otherwise we would have \n\\begin{equation*}\n d(x,y') < l(\\tilde\\sigma) = l(\\sigma_{|[0,d(x,x')]}) + l(\\sigma'_{|[d(x,x'),d(x,x')+d(x',y')]}) = d(x,x') + d(x',y').\n\\end{equation*}\nSince $x'$ lies on a minimal curve between $x$ and $y$, we have $d(x,x') + d(x',y) = d(x,y)$ and we get \n\\begin{equation*}\n  d(x,y') + d(x',y) < d(x,y) + d(x',y')\n\\end{equation*}\nwhich gives a contradiction. It follows that $\\sigma$ and $\\tilde\\sigma$ are unit-speed minimal curves that coincide on the non trivial interval $[0,d(x,x')]$. Since $\\mathbb{H}^n$ is non-branching (see Proposition~\\ref{nonbranching}), this implies that $\\sigma$ and $\\tilde \\sigma$ are sub-arcs of the same minimal curve, namely $\\sigma$ if $d(x,y') \\leq d(x,y)$ and $\\tilde\\sigma$ otherwise, on which all points $x$, $x'$, $y$ and $y'$ lie. And the conclusion follows.\n\\end{proof}\n\nWe denote by $\\Pi_2(\\mu,\\nu)$ the set of transport plans solution to the secondary variational problem:\n\\begin{equation*}\n\\min_{\\gamma \\in \\Pi_1(\\mu,\\nu)} \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)^2\\,d\\gamma(x,y).\n\\end{equation*}\n\nOptimal transport plans selected through the variational approximations to be introduced in Section~\\ref{varapprox} will be solution to this secondary variational problem. The next lemma gives a one-dimensional monotonicity condition along minimal curves used by optimal transport plans in $\\Pi_2(\\mu,\\nu)$. This follows essentially from a constrained version of $d^2$-cyclical monotonicity.\n\n\\begin{lem} \\label{pi2}\nLet $\\gamma\\in \\Pi_2(\\mu,\\nu)$. Then $\\gamma$ is concentrated on a set $\\Gamma$ such that the following holds. For all $(x,y)\\in \\Gamma$ and $(x',y')\\in \\Gamma$ such that $x\\not=y$ and $x\\not=x'$, if $x'$ lies on a minimal curve between $x$ and $y$ then all points $x$, $x'$, $y$ and $y'$ lie on the same minimal curve ordered in that way.\n\\end{lem}\n\nIn other words, there exists a minimal curve $\\sigma:[a,b]\\rightarrow\\mathbb{H}^n$ such that $\\sigma(a) = x$, $\\sigma(t)=y$ for some $t\\in \\,(a,b]$, $\\sigma(s) = x'$ for some $s\\in\\, (a,t]$ and $\\sigma(t') = y'$ for some $t'\\in [t,b]$.\n\n\\begin{proof} First, as a classical fact, one can rephrase the secondary variational problem as a classical Kantorovich transport problem \\eqref{e:MK} between $\\mu$ and $\\nu$ with cost $c(x,y) = \\beta(x,y)$ with \n\\begin{equation*}\n\\beta(x,y) = \\begin{cases}\n              d(x,y)^2 \\quad \\text{if } u(x)-u(y) = d(x,y),\\\\\n              +\\infty \\qquad \\phantom{\\text{if}} \\text{otherwise},\n             \\end{cases}\n\\end{equation*}\nwhere $u\\in \\text{Lip}_1(d)$ is a Kantorovich potential associated to Kantorovich transport problem~\\eqref{e:MK} between $\\mu$ and $\\nu$ with cost $c(x,y) = d(x,y)$ (see Section~\\ref{opttrans} and Theorem~\\ref{1lip_potential} there). Since $\\beta$ is lower semicontinuous and $\\int_{\\mathbb{H}^n\\times \\mathbb{H}^n} \\beta(x,y) \\,d\\gamma(x,y)<+\\infty$ for all $\\gamma \\in \\Pi_2(\\mu,\\nu)$, it follows from Theorem \\ref{ccycl} that any $\\gamma \\in \\Pi_2(\\mu,\\nu)$ is concentrated on a $\\beta$-cyclically monotone set. So, taking into account the fact that $\\Pi_2(\\mu,\\nu)\\subset\\Pi_1(\\mu,\\nu)$, we know that $\\gamma \\in \\Pi_2(\\mu,\\nu)$ is concentrated on a set $\\Gamma$ such that \n\\begin{equation*}\n u(x) - u(y) = d(x,y)\n\\end{equation*}\nfor all $(x,y)\\in \\Gamma$,\n\\begin{equation*}\n \\beta(x,y) +\\beta(x',y') \\leq \\beta(x,y')+\\beta(x',y)\n\\end{equation*}\nfor all $(x,y)\\in \\Gamma$ and $(x',y')\\in \\Gamma$ and the conclusion of Lemma \\ref{pi1.2} holds.\n\nThen let $(x,y)\\in \\Gamma$ and $(x',y')\\in \\Gamma$ be as in the statement. By Lemma \\ref{pi1.2}, the conclusion will follow if we show that $d(x',y) \\leq d(x',y')$. First we check that $\\beta(x',y) = d(x',y)^2$ and $\\beta(x,y') = d(x,y')^2$. We have \n\\begin{equation*}\nu(x) \\leq u(x') + d(x,x') \\leq u(y) + d(x',y) + d(x,x') = d(x,y) + u(y) = u(x)\n\\end{equation*}\nhence all these inequalities are equalities. In particular, we get that $u(x') = u(y) + d(x',y)$ hence $\\beta(x',y) = d(x',y)^2$. We also get that  \n\\begin{equation*}\n u(x) = d(x,x') + u(x') = d(x,x') + u(y') + d(x',y') = u(y') + d(x,y')\n\\end{equation*}\nhence $\\beta(x,y') = d(x,y')^2$. If $d(x',y')< d(x',y)$, we  get\n\\begin{equation*}\n\\begin{split}\n \\beta(x,y') + &\\beta(x',y) - \\beta(x',y') - \\beta(x,y)\\\\\n&= d(x,y')^2  + d(x',y)^2 - d(x',y')^2 - d(x,y)^2 \\\\\n&= (d(x,x')+d(x',y'))^2 + d(x',y)^2 - d(x',y')^2 - (d(x,x')+d(x',y))^2\\\\\n&= 2 \\, d(x,x') (d(x',y') - d(x',y))<0\n\\end{split}\n\\end{equation*}\nwhich gives a contradiction.\n\\end{proof}\n\n\n\\section{Variational approximations} \\label{varapprox}\n\nWe introduce variational approximations in the spirit of \\cite{ap} (see also \\cite{cfm}, \\cite{akp}) by rephrasing in our geometrical context the variational approximations considered recently in \\cite{sant}. This approximation procedure will be used to select optimal transport plans that will be eventually proved to be induced by transport maps.\n\nLet $\\mu$, $\\nu \\in \\mathcal P_c(\\mathbb{H}^n)$ be fixed. Let $K$ be a compact subset of $\\mathbb{H}^n$ such that $\\operatorname{spt}{\\mu}\\cup\\operatorname{spt}{\\nu}\\subset K$ and set\n\\begin{equation*}\n \\Pi := \\{\\gamma \\in \\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n);\\, (\\pi_1)_\\sharp\\gamma=\\mu,\\, \\operatorname{spt}{(\\pi_2)_\\sharp\\gamma} \\subset K\\}.\n\\end{equation*}\nFor $\\varepsilon>0$ fixed and $\\gamma \\in \\Pi$, we set\n\\begin{multline*}\n C_\\varepsilon(\\gamma) := \n\\frac{1}{\\varepsilon}\\, W_1((\\pi_2)_\\sharp\\gamma,\\nu) + \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\gamma(x,y) \\\\ +  \\varepsilon \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)^2\\,d\\gamma(x,y) + \\varepsilon^{6n+8} \\operatorname{card}{(\\operatorname{spt}{(\\pi_2)_\\sharp\\gamma})}\n\\end{multline*}\nand consider the family of minimization problems:\n\\begin{equation} \\label{e:varapprox}\n \\min\\{ C_\\varepsilon(\\gamma);\\, \\gamma \\in \\Pi\\}. \\tag{$P_\\varepsilon$}\n\\end{equation}\nHere $W_1$ denotes the 1-Wasserstein distance defined for any two probability measures $\\mu_1$, $\\mu_2 \\in \\mathcal P(\\mathbb{H}^n)$ by \n\\begin{equation*}\n W_1(\\mu_1,\\mu_2) := \\min_{\\gamma \\in \\Pi(\\mu_1,\\mu_2)} \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\gamma(x,y).\n\\end{equation*}\nFirst we note that \\eqref{e:varapprox} always admits solutions.\n\n\\begin{thm}\n For any $\\varepsilon>0$, the problem \\eqref{e:varapprox} admits at least one solution and $\\min\\{ C_\\varepsilon(\\gamma);\\, \\gamma \\in \\Pi\\}<+\\infty$.\n\\end{thm}\n\n\\begin{proof}\nFirst note that since $K$ is compact, $C_\\varepsilon(\\gamma)<+\\infty$ for any $\\gamma\\in \\Pi$ such that $(\\pi_2)_\\sharp\\gamma$ is finitely atomic. Next the existence of solutions to \\eqref{e:varapprox} follows from the weak compactness of $\\Pi$, the lower semicontinuity of the three first terms to be minimized and the Kuratowski convergence of the supports of weakly converging probability measures (see \\cite[Chapter~5]{ags}).\n\\end{proof}\n\nNext, weak limits of solutions to \\eqref{e:varapprox} are optimal transport plans that are solutions to the secondary variational problem introduced in Section~\\ref{sect:optplanning} to which we refer for the definition of $\\Pi_2(\\mu,\\nu)$. Modulo minor modifications due to our geometrical context, this can be proved with the same arguments as those given in \\cite{sant}. \n\n\\begin{lem} \\label{optpi2}\n Let $\\varepsilon_k$ be a sequence converging to 0 and $\\gamma_{\\varepsilon_k}$ a sequence of solutions to $(P_{\\varepsilon_k})$ which is weakly converging to some $\\gamma\\in \\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n)$. Then $\\gamma \\in \\Pi_2(\\mu,\\nu)$.\n\\end{lem}\n\n\\begin{proof} First we note that for any $m\\geq 1$, one can find a finite set $F_m\\subset K$ such that $\\operatorname{card} F_m \\leq C\\, m^{2n+2}$ for some constant $C>0$ which depends only on $n$ and $\\operatorname{diam} K$ and a Borel map $p_m:K \\rightarrow F_m$ such that\n\\begin{equation*}\n d(p_m(x),x) < 1\/m  \n\\end{equation*}\nfor all $x\\in K$. Indeed choose $x_1 \\in K$. For $i\\geq 2$, choose by induction $x_i \\in K\\setminus \\cup_{j<i} B(x_j,1\/m)$ as long as $K\\setminus \\cup_{j<i} B(x_j,1\/m) \\not=\\emptyset$. Let $F_m$ denote the set of all these points. The balls $B(x_i,1\/(2m))$ are mutually disjoint and $\\cup_{i} B(x_i,(1\/(2m)) \\subset B(x_1,\\operatorname{diam} K + 1)$. Remembering \\eqref{e:measball}, it follows that\n\\begin{equation*}\n\\begin{split}\n c_n (2m)^{-2n-2} \\operatorname{card} F&\\leq \\mathcal{L}^{2n+1}(\\cup_i B(x_i,1\/(2m)))\\\\\n&\\leq \\mathcal{L}^{2n+1}(B(x_1,\\operatorname{diam} K +1)) = c_n (\\operatorname{diam} K + 1)^{2n+2}\n\\end{split}\n\\end{equation*}\nfor any finite subset $F\\subset F_m$, hence $F_m$ is a finite set with $\\operatorname{card} F_m \\leq C\\, m^{2n+2}$ where $C$ depends only on $n$ and $\\operatorname{diam} K$. Next, by construction, for any $x \\in K$, there exists a unique $x_i\\in F_m$ such that $x \\in B(x_i,1\/m)\\setminus \\cup_{j<i} B(x_j,1\/m)$ and we then set $p_m(x) := x_i$. \n\nThe proof of the lemma can now be completed following the same arguments as those in \\cite{sant}. For sake of completeness, we sketch these arguments below. Let $\\gamma_{\\varepsilon_k}$ and $\\gamma$ be as in the statement. For $m\\geq 1$, one sets $\\nu_m:= (p_m)_\\sharp\\nu$. Note that by construction of $F_m$ and $p_m$, one has $\\operatorname{card}{(\\operatorname{spt}{\\nu_m})} \\leq  C\\, m^{2n+2}$ and $W_1(\\nu_m,\\nu) \\leq 1\/m$. To check that $\\gamma \\in \\Pi(\\mu,\\nu)$, one takes some $\\gamma_m \\in \\Pi(\\mu,\\nu_m)\\subset \\Pi$ and uses the optimality of $\\gamma_{\\varepsilon_k}$ which implies \n\\begin{equation*}\n\\begin{split}\n W_1((\\pi_2)_\\sharp\\gamma_{\\varepsilon_k},\\nu) &\\leq \\varepsilon_k \\, C_{\\varepsilon_k}(\\gamma_m) \\\\\n&\\leq \\dfrac{1}{m} + \\varepsilon_k \\operatorname{diam}(K) + \\varepsilon_k^2 \\operatorname{diam}(K)^2 + C \\varepsilon_k^{6n+9} m^{2n+2}.\n\\end{split}\n\\end{equation*}\nThen one lets $\\varepsilon_k \\rightarrow 0$ with $m\\geq 1$ fixed and then $m \\rightarrow +\\infty$ to get that $(\\pi_2)_\\sharp\\gamma_{\\varepsilon_k}$ converges weakly to $\\nu$. Since it also converges weakly to $(\\pi_2)_\\sharp\\gamma$, it follows that $\\gamma \\in \\Pi(\\mu,\\nu)$. \n\nTo check that $\\gamma \\in \\Pi_1(\\mu,\\nu)$, one takes $\\overline\\gamma \\in \\Pi_1(\\mu,\\nu)$ and sets $\\overline\\gamma_m := (Id,p_m)_\\sharp\\overline\\gamma \\in \\Pi(\\mu,\\nu_m)$ where $(Id,p_m)(x,y) = (x,p_m(y))$. By optimality of $\\gamma_{\\varepsilon_k}$, one has \n\\begin{equation*}\n\\begin{split}\n \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)&\\,d\\gamma_{\\varepsilon_k}(x,y) \\leq  C_{\\varepsilon_k}(\\overline\\gamma_m)\\\\\n&\\leq \\dfrac{1}{m \\,\\varepsilon_k} + \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\overline\\gamma_m(x,y) + \\varepsilon_k \\operatorname{diam}(K)^2 + C \\varepsilon_k^{6n+8} m^{2n+2}.\n\\end{split}\n\\end{equation*}\nChoosing $m$ of the order of $\\varepsilon_k^{-2}$ and letting $\\varepsilon_k \\rightarrow 0$, one gets \n\\begin{equation*}\n \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\gamma(x,y) \\leq  \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\overline\\gamma(x,y)\n\\end{equation*}\nhence $\\gamma \\in \\Pi_1(\\mu,\\nu)$. \n\nFinally, to check that $\\gamma \\in \\Pi_2(\\mu,\\nu)$, one uses once again the optimality of $\\gamma_{\\varepsilon_k}$ in the following way,\n\\begin{multline*}\n  W_1((\\pi_2)_\\sharp\\gamma_{\\varepsilon_k},\\nu) + \\varepsilon_k \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\gamma_{\\varepsilon_k}(x,y) + \\varepsilon_k^2 \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)^2\\,d\\gamma_{\\varepsilon_k}(x,y) \\\\\\leq \\varepsilon_k \\, C_{\\varepsilon_k}(\\overline\\gamma_m) \n\\end{multline*}\nOn the other hand, one has \n\\begin{gather*}\n\\begin{split}\n W_1(\\mu,\\nu) &\\leq W_1(\\mu,(\\pi_2)_\\sharp\\gamma_{\\varepsilon_k}) + W_1((\\pi_2)_\\sharp\\gamma_{\\varepsilon_k},\\nu)\\\\\n& \\leq \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\gamma_{\\varepsilon_k}(x,y)  + W_1((\\pi_2)_\\sharp\\gamma_{\\varepsilon_k},\\nu)\n\\end{split}\\\\\n\\intertext{and}\n\\begin{split}\n\\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\overline\\gamma_m(x,y) &= \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,p_m(y))\\,d\\overline\\gamma(x,y)\\\\\n&\\leq \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)\\,d\\overline\\gamma(x,y) + \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(y,p_m(y))\\,d\\overline\\gamma(x,y) \\\\\n&\\leq  W_1(\\mu,\\nu) + \\frac{1}{m}.\n\\end{split}\n\\end{gather*}\nIt follows that \n\\begin{multline*}\n \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)^2\\,d\\gamma_{\\varepsilon_k}(x,y) \\\\ \\leq \\frac{1}{m\\, \\varepsilon_k^2} + \\frac{1}{m\\, \\varepsilon_k} + \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)^2\\,d\\overline\\gamma_m(x,y) + C \\varepsilon_k^{6n+7} m^{2n+2}\n\\end{multline*}\nprovided $\\varepsilon_k\\leq 1$. Choosing $m$ of the order of $\\varepsilon_k^{-3}$ and letting $\\varepsilon_k \\rightarrow 0$, one gets\n\\begin{equation*}\n \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)^2\\,d\\gamma(x,y) \\leq \\int_{\\mathbb{H}^n\\times\\mathbb{H}^n} d(x,y)^2\\,d\\overline\\gamma(x,y).\n\\end{equation*}\nSince $\\overline\\gamma \\in \\Pi_1(\\mu,\\nu)$ was arbitrary, it follows that $\\gamma \\in \\Pi_2(\\mu,\\nu)$.\n\\end{proof}\n\\medskip\n\nThe rest of this section is devoted to the study of the solutions to ($P_\\varepsilon$). We fix $\\varepsilon>0$ and set \n\\begin{equation*}\n c_\\varepsilon (x,y) := d(x,y) + \\varepsilon\\, d(x,y)^2.\n\\end{equation*}\nWe first recall the following classical fact.\n\n\\begin{lem} \\label{restrictions}\nLet $\\gamma_\\varepsilon$ be a solution to ($P_\\varepsilon$). Then for any Borel set $U\\subset\\mathbb{H}^n\\times\\mathbb{H}^n$, $(\\pi_2)_\\sharp(\\gamma_\\varepsilon\\lfloor U)$ is finitely atomic and  $\\gamma_\\varepsilon\\lfloor U$ is a solution to Kantorovich transport problem \\eqref{e:MK} between $(\\pi_1)_\\sharp (\\gamma_\\varepsilon\\lfloor U)$ and $(\\pi_2)_\\sharp (\\gamma_\\varepsilon\\lfloor U)$ with  cost $c_\\varepsilon$.\n\\end{lem}\n\n\\begin{proof}\n The fact that $(\\pi_2)_\\sharp(\\gamma_\\varepsilon\\lfloor U)$ is finitely atomic obviously follows from the fact that $C_\\varepsilon(\\gamma_\\varepsilon) = \\min\\{ C_\\varepsilon(\\gamma);\\, \\gamma \\in \\Pi\\}<+\\infty$. Next it is also immediate that $\\gamma_\\varepsilon$ is a solution to Kantorovich transport problem \\eqref{e:MK} between $\\mu$ and $(\\pi_2)_\\sharp (\\gamma_\\varepsilon)$ with cost $c_\\varepsilon$. Then as a classical fact, the claim follows from the linearity of the functional to be minimized with respect to the transport plan. If $\\gamma \\in \\Pi((\\pi_1)_\\sharp (\\gamma_\\varepsilon\\lfloor U),(\\pi_2)_\\sharp(\\gamma_\\varepsilon\\lfloor U))$, one indeed simply compare $C_\\varepsilon(\\gamma_\\varepsilon)$ with $C_\\varepsilon(\\hat\\gamma)$ where $ \\hat\\gamma = \\gamma_\\varepsilon\\lfloor (\\mathbb{H}^n\\times\\mathbb{H}^n)\\setminus U + \\gamma \\in \\Pi(\\mu,(\\pi_2)_\\sharp(\\gamma_\\varepsilon))$ to get the conclusion.\n\\end{proof}\n\nNext in this section we consider interpolations between two measures $\\overline\\mu$, $\\overline \\nu\\in \\mathcal P_c(\\mathbb{H}^n)$ that are constructed from a transport plan solution to Kantorovich transport problem \\eqref{e:MK} between these two measures with cost $c_\\varepsilon$. We prove absolute continuity and, more importantly, $L^\\infty$-estimates on the density with respect to $\\mathcal{L}^{2n+1}$ of these interpolations whenever $\\overline\\mu \\ll \\mathcal{L}^{2n+1}$ and $\\overline \\nu$ is finitely atomic, see Proposition~\\ref{e:Linftydensityestimates}. We divide the arguments into several steps. First we prove that any solution to this Kantorovich transport problem is induced by a transport.\n\n\\begin{thm}  \\label{existenceceps}\n Let $\\overline\\mu$, $\\overline \\nu\\in \\mathcal P_c(\\mathbb{H}^n)$ be fixed. Assume that $\\overline\\mu \\ll \\mathcal{L}^{2n+1}$ and that $\\overline \\nu$ is finitely atomic. Then any solution to Kantorovich transport problem \\eqref{e:MK} between $\\overline\\mu$ and $\\overline \\nu$ with  cost $c_\\varepsilon$ is induced by a transport. In particular there exists a unique optimal transport map solution to the transport problem \\eqref{e:M} between $\\overline\\mu$ and $\\overline \\nu$ with cost $c_\\varepsilon$.\n\\end{thm}\n\n\\begin{proof} Let $\\psi$ be a Kantorovich potential for Kantorovich transport problem \\eqref{e:MK} between $\\overline\\mu$ and $\\overline \\nu$ with  cost $c_\\varepsilon$ given by Theorem~\\ref{duality}. Let $\\{y_i\\}_{i=1}^k$ denote the atoms of $\\overline \\nu$. We prove that for $\\mathcal{L}^{2n+1}$-a.e. $x\\in\\mathbb{H}^n$, there is at most one point $y_i$ for some $i\\in\\{1,\\dots,k\\}$ such that\n\\begin{equation*} \n \\psi(x) + \\psi^c(y_i) = c_\\varepsilon (x,y_i).\n\\end{equation*}\nSince $\\overline\\mu \\ll \\mathcal{L}^{2n+1}$, it will follow that any transport plan solution to Kantorovich transport problem \\eqref{e:MK} between $\\overline\\mu$ and $\\overline \\nu$ with  cost $c_\\varepsilon$ is concentrated on a $\\overline\\mu$-measurable graph and hence induced by a transport. This implies in turn existence and uniqueness of the optimal transport map solution to the transport problem \\eqref{e:M} between $\\overline\\mu$ and $\\overline \\nu$ with cost $c_\\varepsilon$ (see Theorem~\\ref{plan-transport}). \n\nFor $i\\not= j$, set $h_{ij}(x) := c_\\varepsilon (x,y_i) - c_\\varepsilon (x,y_j) + \\psi^c(y_j) - \\psi^c(y_i)$. It follows from Lemma~\\ref{prop-distcc} that $h_{ij}$ is of class $C^\\infty$ on the open set $\\mathbb{H}^n\\setminus(L_{y_i} \\cup L_{y_j})$ with $\\nabla h_{ij} \\not= 0$. Indeed assume on the contrary that $\\nabla h_{ij}(x) =0$ for some $x\\in \\mathbb{H}^n\\setminus(L_{y_i} \\cup L_{y_j})$. Then, differentiating along the horizontal vector fields $X_j$ and $Y_j$, we would have\n\\begin{equation*} \n \\nabla_H d_{y_i} (x) \\,(1+2\\varepsilon\\, d_{y_i}(x)) =  \\nabla_H d_{y_j} (x) \\,(1+2\\varepsilon \\,d_{y_j}(x)).\n\\end{equation*}\nSince $|\\nabla_H d_{y_i} (x) | = |\\nabla_H d_{y_j} (x) |$ (see Lemma~\\ref{prop-distcc}(i)), this would imply that $d_{y_i}(x) = d_{y_j}(x)$ and in turn that $\\nabla_H d_{y_i} (x) = \\nabla_H d_{y_j} (x)$. Since we also have by assumption $\\partial_t d_{y_i} (x) = \\partial_t d_{y_j} (x)$, Lemma~\\ref{prop-distcc}(ii) would give $y_i=y_j$. It follows that the set $\\{x\\in\\mathbb{H}^n\\setminus(L_{y_i} \\cup L_{y_j});\\, h_{ij}(x) =0\\}$ is a $C^\\infty$-smooth submanifold of dimension $2n$ in $\\mathbb{R}^{2n+1}$ and hence has Lebesgue measure 0. Since $\\mathcal{L}^{2n+1}(L_{y_i}) =0$, it follows that \n\\begin{equation*} \n \\mathcal{L}^{2n+1} (\\bigcup_{i\\not=j} \\{x\\in\\mathbb{H}^n;\\,h_{ij}(x) =0\\})  = 0\n\\end{equation*}\nand the claim follows.\n\\end{proof}\n\nIf $T:\\mathbb{H}^n \\rightarrow \\mathbb{H}^n$, we set $T_t = e_t \\circ S \\circ (I \\otimes T)$, i.e., $T_t(x)$ is the point lying at distance $t \\, d(x,T(x))$ from $x$ on the selected minimal curve $S(x,T(x))$ between $x$ and $T(x)$ (see Subsection~\\ref{sect-interpolation} for the definition of $S$ and $e_t$).\n\n\\begin{prop} \\label{injectivity} \\cite[Chapter 7]{villani}\n Let $\\overline\\mu$, $\\overline \\nu\\in \\mathcal P_c(\\mathbb{H}^n)$ be fixed such that $\\overline\\mu \\ll \\mathcal{L}^{2n+1}$ and $\\overline \\nu$ is finitely atomic. Let $T^\\varepsilon$ be the optimal transport map solution to the transport problem \\eqref{e:M} between $\\overline\\mu$ and $\\overline \\nu$ with cost $c_\\varepsilon$. Then there exists a $\\overline\\mu$ - measurable set $A$ such that $\\overline\\mu(A) = 1$ and such that for each $t\\in\\, [0,1)$, $T^\\varepsilon_t\\lfloor_A$ is injective.\n\\end{prop}\n\nThe cost $c_\\varepsilon$ can be recovered as coming from a so-called coercive Lagrangian action. Since $(\\mathbb{H}^n,d)$ is non-branching, the proposition essentially follows from~\\cite[Chapter 7, Theorem 7.30]{villani}. However one does not need the full strength of the theory developed in~\\cite[Chapter 7]{villani} to get the conclusion of Proposition~\\eqref{injectivity} and we sketch below the arguments for the reader's convenience. \n\n\\begin{proof} Let $0\\leq s<t \\leq 1$ and $x$, $y\\in \\mathbb{H}^n$. Set \n\\begin{equation*} \nc_\\varepsilon^{s,t} (x,y) = d(x,y) + \\varepsilon \\, \\dfrac{d(x,y)^2}{t-s}.\n\\end{equation*}\nThe space $(\\mathbb{H}^n,d)$ being a geodesic space and $u \\mapsto u + \\varepsilon\\,u^2$ being strictly increasing and strictly convex on $[0,+\\infty )$, one has \n\\begin{equation*} \n c_\\varepsilon (x,y) \\leq c_\\varepsilon^{0,t} (x,z) + c_\\varepsilon^{t,1} (z,y)\n\\end{equation*}\nfor all $x$, $y$, $z\\in \\mathbb{H}^n$ and $t\\in (0,1)$, with equality if and only if any curve in $C([0,1],\\mathbb{H}^n)$ obtained by concatenation of a minimal curve $\\sigma_{x,z}:[0,t]\\rightarrow\\mathbb{H}^n$ between $x$ and $z$ and a minimal curve $\\sigma_{z,y}:[t,1]\\rightarrow\\mathbb{H}^n$ between $z$ and $y$ is a minimal curve between $x$ and $y$.\n\nOn the other hand, by $c_\\varepsilon$-cyclical monotonicity, one knows that there exists a $\\overline\\mu$ - measurable set $A$ such that $\\overline\\mu(A) =1$ and \n\\begin{equation*} \n c_\\varepsilon(x,T^\\varepsilon(x)) + c_\\varepsilon(\\tilde x,T^\\varepsilon(\\tilde x)) \\leq c_\\varepsilon(x,T^\\varepsilon(\\tilde x)) + c_\\varepsilon(\\tilde x,T^\\varepsilon(x))\n\\end{equation*}\nfor all $x$, $\\tilde x \\in A$ (see Theorem~\\ref{ccycl}).\n\nNow let $t\\in (0,1)$ be fixed and let $x$, $\\tilde x \\in A$. Assume that $T^\\varepsilon_t(x) = T^\\varepsilon_t(\\tilde x)$. Then \n\\begin{equation} \n c_\\varepsilon(x,T^\\varepsilon(\\tilde x)) \\leq c_\\varepsilon^{0,t} (x,z) + c_\\varepsilon^{t,1} (z,T^\\varepsilon(\\tilde x)),\\label{e:injectivity}\n\\end{equation}\nand similarly,\n\\begin{equation*} \nc_\\varepsilon(\\tilde x,T^\\varepsilon(x)) \\leq c_\\varepsilon^{0,t} (\\tilde x,z) + c_\\varepsilon^{t,1} (z,T^\\varepsilon(x))\n\\end{equation*}\nwhere $z = T^\\varepsilon_t(x) = T^\\varepsilon_t(\\tilde x)$. It follows that \n\\begin{equation*} \n\\begin{split}\nc_\\varepsilon(x,& T^\\varepsilon(x)) + c_\\varepsilon(\\tilde x,T^\\varepsilon(\\tilde x))\\\\ &\\leq c_\\varepsilon(x,T^\\varepsilon(\\tilde x)) + c_\\varepsilon(\\tilde x,T^\\varepsilon(x))\\\\\n&\\leq c_\\varepsilon^{0,t} (x,T^\\varepsilon_t(x)) + c_\\varepsilon^{t,1} (T^\\varepsilon_t(x),T^\\varepsilon(x)) + c_\\varepsilon^{0,t} (\\tilde x,T^\\varepsilon_t(\\tilde x)) + c_\\varepsilon^{t,1} (T^\\varepsilon_t(\\tilde x),T^\\varepsilon(\\tilde x))\\\\\n&= c_\\varepsilon(x,T^\\varepsilon(x)) + c_\\varepsilon(\\tilde x,T^\\varepsilon(\\tilde x)).\n\\end{split}\n\\end{equation*}\nHence equality has to hold in all these inequalities. In particular equality holds in \\eqref{e:injectivity}. It follows that the curve obtained by concatenation of the minimal curve $s\\in [0,t] \\mapsto e_s(S(x,T^\\varepsilon(x)))$ between $x$ and $z$ with the minimal curve $s \\in [t,1] \\mapsto e_s(S(\\tilde x,T^\\varepsilon(\\tilde x)))$ between $z$ and $T^\\varepsilon(\\tilde x)$ is a minimal curve. Since this curve coincide with the minimal curve $\\sigma: s\\in [0,1] \\mapsto e_s(S(x,T^\\varepsilon(x)))$ on the non trivial interval $[0,t]$ and $\\mathbb{H}^n$ is non-branching (see Proposition~\\ref{nonbranching}), we get that it coincides with $\\sigma$ on the whole interval $[0,1]$. Similarly, it coincides with the minimal curve $\\tilde \\sigma: s\\in [0,1] \\mapsto e_s(S(\\tilde x,T^\\varepsilon(\\tilde x)))$ on the whole interval $[0,1]$. Hence $\\sigma = \\tilde \\sigma$ and in particular $x=\\sigma(0) = \\tilde \\sigma (0) = \\tilde x$.\n\\end{proof}\n\nWe turn now to the main estimate that will lead to Proposition~\\ref{e:Linftydensityestimates}.\n\n\\begin{prop} \\label{estimsup}\n Let $\\overline\\mu \\in \\mathcal P(\\mathbb{H}^n)$, $\\overline\\mu \\ll \\mathcal{L}^{2n+1}$, $\\overline\\mu = \\rho \\,d\\mathcal{L}^{2n+1}$, and $T:\\mathbb{H}^n\\rightarrow\\mathbb{H}^n$ be a $\\overline\\mu$ - measurable map such that $T_\\sharp\\overline\\mu$ is finitely atomic. Let $t\\in\\,(0,1)$ and set $\\overline\\mu_t := T_{t\\,\\sharp}\\overline\\mu$. Assume that there exists a $\\overline\\mu$ - measurable set $A$ such that $\\overline\\mu(A) = 1$ and $T_t\\lfloor_A$ is injective. Then $\\overline\\mu_t \\ll \\mathcal{L}^{2n+1}$, $\\overline\\mu_t = \\rho_t \\,d\\mathcal{L}^{2n+1}$ with \n\\begin{equation*}\n \\rho_t \\leq \\frac{{\\rm 1\\mskip-4mu l}_{T_t(A)}}{(1-t)^{2n+3}} \\,\\rho \\circ T_t^{-1}\\lfloor_{T_t(A)} \\qquad \\mathcal{L}^{2n+1}-a.e.\n\\end{equation*}\n\\end{prop}\n\nArguments for the proof of this proposition can be found in \\cite[Section~3]{figjuil} even-though not explicitly stated in the same way in that paper. They rely on the following estimate:\n\\begin{equation*}\n \\mathcal{L}^{2n+1}(E) \\leq \\frac{1}{(1-t)^{2n+3}} \\,\\mathcal{L}^{2n+1}((e_t \\circ S)(E,y))\n\\end{equation*}\nfor any $y\\in\\mathbb{H}^n$ and $E\\subset \\mathbb{H}^n$ which is proved in \\cite[Section~2]{juillet} and which roughly means that $(\\mathbb{H}^n,d,\\mathcal{L}^{2n+1})$ satisfies a  so-called Measure Contraction Property. We detail the proof below for the reader's convenience.\n\n\\begin{proof} \n Let $\\{y_i\\}_{i=1}^k$ denote the atoms of $T_\\sharp\\overline\\mu$. Set $A_i = T^{-1}(\\{y_i\\}) \\cap A$ and $\\hat A = \\cup_i A_i$. The sets $A_i$ are mutually disjointed and $\\overline\\mu(\\hat A)=1$ by hypothesis. For any $x\\in A_i$, $T_t(x) = (e_t \\circ S) (x,y_i)$, hence\n\\begin{equation*}\n \\mathcal{L}^{2n+1}(E) \\leq \\frac{1}{(1-t)^{2n+3}} \\,\\mathcal{L}^{2n+1}(T_t(E))\n\\end{equation*}\nfor any $E\\subset A_i$. Next if $E\\subset \\hat A$, writing $E = \\cup_i (E\\cap A_i)$ where the sets $A_i$ are mutually disjointed and remembering that $T_t$ is injective on $\\hat A\\subset A$ by hypothesis, one gets\n\\begin{equation*}\n  \\mathcal{L}^{2n+1}(E) \\leq \\frac{1}{(1-t)^{2n+3}}\\, \\mathcal{L}^{2n+1}(T_t(E)).\n\\end{equation*}\nIt follows that for any $F\\subset \\mathbb{H}^n$,\n\\begin{equation*} \\label{e:1}\n \\mathcal{L}^{2n+1}(T_t^{-1}(F)\\cap \\hat A) \\leq \\frac{1}{(1-t)^{2n+3}} \\mathcal{L}^{2n+1}(F\\cap T_t(\\hat A)).\n\\end{equation*}\n\nAssume that $F\\subset \\mathbb{H}^n$ is such that $\\mathcal{L}^{2n+1}(F)=0$. We get $\\mathcal{L}^{2n+1}(T_t^{-1}(F)\\cap \\hat A) = 0$ from the previous inequality. On the other hand $\\overline\\mu_t(F) = \\overline \\mu (T_t^{-1}(F)) = \\overline \\mu (T_t^{-1}(F)\\cap \\hat A)$ hence $\\overline\\mu_t(F)=0$ since $\\overline\\mu\\ll\\mathcal{L}^{2n+1}$ and it follows that $\\overline\\mu_t \\ll \\mathcal{L}^{2n+1}$. \n\nNext, to prove the estimate on the density of $\\overline\\mu_t$ with respect to $\\mathcal{L}^{2n+1}$, we note that the inequality above implies that \n\\begin{equation*}\n \\int_{\\hat A} g(T_t) \\, d\\mathcal{L}^{2n+1} \\leq \\frac{1}{(1-t)^{2n+3}} \\int_{T_t(\\hat A)} g \\, d\\mathcal{L}^{2n+1}\n\\end{equation*}\nfor any non negative measurable map $g:\\mathbb{H}^n\\rightarrow [0,+\\infty]$. Let $h:\\mathbb{H}^n\\rightarrow [0,+\\infty]$ be a non negative measurable map and set \n\\begin{equation*}\n g(x) = {\\rm 1\\mskip-4mu l}_{T_t(\\hat A)}(x) \\, h(x) \\, \\rho(T_t^{-1}\\lfloor_{T_t(\\hat A)}(x)).\n\\end{equation*}\nThen \n\\begin{equation*}\n \\int_{\\hat A} g(T_t(x)) \\, d\\mathcal{L}^{2n+1}(x) \\leq \\frac{1}{(1-t)^{2n+3}} \\int_{T_t(\\hat A)} h(x) \\,\\rho(T_t^{-1}\\lfloor_{T_t(\\hat A)}(x)) \\, d\\mathcal{L}^{2n+1}(x)\n\\end{equation*}\nOn the other hand\n\\begin{equation*}\n \\int_{\\hat A}  g(T_t) \\, d\\mathcal{L}^{2n+1} = \\int_{\\hat A} h(T_t)\\, \\rho \\, d\\mathcal{L}^{2n+1} = \\int_{\\mathbb{H}^n} h(T_t)\\, d\\overline\\mu = \\int_{\\mathbb{H}^n} h\\, d\\overline\\mu_t,\n\\end{equation*}\nhence\n\\begin{equation*}\n \\int_{\\mathbb{H}^n} h(x)\\, d\\overline\\mu_t(x) \\leq \\frac{1}{(1-t)^{2n+3}} \\int_{T_t(\\hat A)} h(x) \\, \\rho(T_t^{-1}\\lfloor_{T_t(\\hat A)}(x)) \\, d\\mathcal{L}^{2n+1}(x).\n\\end{equation*}\nRemembering that $\\hat A \\subset A$, this concludes the proof.\n\\end{proof}\n\nFinally, combining Theorem~\\ref{existenceceps}, Propositions~\\ref{injectivity} and \\ref{estimsup}, we get the following proposition which gives the absolute continuity of the interpolations together with an $L^\\infty$-estimate on their density. Note that if $\\gamma_\\varepsilon$ is the transport plan solution to Kantorovich transport problem \\eqref{e:MK} between $\\overline\\mu$ and $\\overline \\nu$ with cost $c_\\varepsilon$ and $T^\\varepsilon$ the optimal transport map solution to the transport problem \\eqref{e:M} between $\\overline\\mu$ and $\\overline \\nu$ with cost $c_\\varepsilon$, which hence induces $\\gamma_\\varepsilon$, then $(e_t \\circ S)_\\sharp\\gamma_\\varepsilon=T_{t\\,\\sharp}^\\varepsilon\\overline\\mu$.\n\n\\begin{prop} \\label{e:Linftydensityestimates}\n Let $\\overline\\mu$, $\\overline \\nu\\in \\mathcal P_c(\\mathbb{H}^n)$ be fixed. Assume that $\\overline\\mu \\ll \\mathcal{L}^{2n+1}$ with $\\overline\\mu = \\rho \\,d\\mathcal{L}^{2n+1}$ and $\\overline \\nu$ is finitely atomic. Let $\\gamma_\\varepsilon$ be the transport plan solution to Kantorovich transport problem \\eqref{e:MK} between $\\overline\\mu$ and $\\overline \\nu$ with cost $c_\\varepsilon$. Then for any $t\\in [0,1)$, the interpolation $(e_t \\circ S)_\\sharp\\gamma_\\varepsilon$ is absolutely continuous with respect to $\\mathcal{L}^{2n+1}$, $(e_t \\circ S)_\\sharp\\gamma_\\varepsilon = \\rho_t^\\varepsilon \\,d\\mathcal{L}^{2n+1}$, and one has \n\\begin{equation*}\n \\|\\rho_t^\\varepsilon\\|_{L^\\infty} \\leq \\frac{1}{(1-t)^{2n+3}}\\, \\|\\rho\\|_{L^\\infty}.\n\\end{equation*}\n\\end{prop}\n\n\n\n\n\\section{Properties of measures $\\gamma\\in \\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n)$ with $(\\pi_1)_\\sharp\\gamma \\ll \\mathcal{L}^{2n+1}$} \\label{lebpoints}\n\nThis section is independent of the transport problem. We state some properties of measures in $\\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n)$ with first marginal absolutely continuous with respect to $\\mathcal{L}^{2n+1}$. These properties are essential steps in the strategy adopted here to solve Monge's transport problem. They are the exact counterpart in the framework of $(\\mathbb{H}^n, d, \\mathcal{L}^{2n+1})$ of similar properties proved in~\\cite{champion-dePascale} in $\\mathbb{R}^n$. These properties hold actually true in more general settings, for instance in any separable doubling metric measure space.\n\nWe first recall some facts about Lebesgue points of Borel functions and density of absolutely continuous measures. Since the measure $\\mathcal{L}^{2n+1}$ is a doubling measure on $(\\mathbb{H}^n,d)$, see \\eqref{e:measball}, if $\\rho:\\mathbb{H}^n \\rightarrow [0,+\\infty]$ is a $\\mathcal{L}^{2n+1}$-locally summable Borel function then for $\\mathcal{L}^{2n+1}$-a.e. $x\\in \\mathbb{H}^n$, one has \n\\begin{equation} \\label{e:lebpoint}\n \\lim_{r\\rightarrow 0} \\dfrac{1}{\\mathcal{L}^{2n+1}(B(x,r))} \\, \\int_{B(x,r)} |\\rho(y) - \\rho(x)|\\,d\\mathcal{L}^{2n+1}(y) = 0,\n\\end{equation}\nsee e.g. \\cite{heinonen}. A point $x\\in \\mathbb{H}^n$ where \\eqref{e:lebpoint} holds is called a Lebesgue point of $\\rho$ and we denote by $\\operatorname{Leb} \\rho$ the set of all Lebesgue points of $\\rho$. \n\nIn rest of this paper, and especially in the lemma to follow, it will be technically convenient to consider the density $\\rho$ of an absolutely continuous measure $\\mu \\ll \\mathcal{L}^{2n+1}$ as a $\\mathcal{L}^{2n+1}$-summable Borel function, i.e., a function well-defined everywhere, so that one can speak about its Lebesgue points and its value at any arbitrary point without any ambiguity. If $\\mu\\in \\mathcal P(\\mathbb{H}^n)$, we set \n\\begin{equation} \\label{e:density}\n \\rho(x) := \\limsup_{r\\rightarrow 0} \\dfrac{\\mu(B(x,r))}{\\mathcal{L}^{2n+1}(B(x,r))}\\,.\n\\end{equation}\nThis map $\\rho:\\mathbb{H}^n \\rightarrow [0,+\\infty]$ is a Borel map and if $\\mu\\ll \\mathcal{L}^{2n+1}$ then $\\mu = \\rho \\,d\\mathcal{L}^{2n+1}$. By a slight abuse of terminology, when speaking about the density of an absolutely continuous measure $\\mu\\in \\mathcal P(\\mathbb{H}^n)$ with respect to $\\mathcal{L}^{2n+1}$, we will thus always refer in the following to the Borel function $\\rho$ defined above.\n\nThe next lemma will be an essential ingredient in the proof of Lemma~\\ref{mainlemma}. \n\n\\begin{lem} \\label{dens1}\n Let $\\gamma\\in \\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n)$ be such that $(\\pi_1)_\\sharp\\gamma \\ll \\mathcal{L}^{2n+1}$. Then $\\gamma$ is concentrated on a set $\\Gamma$ such that, for all $(x,y)\\in  \\Gamma$ and all $r>0$, there exist $y'\\in\\mathbb{H}^n$ and $r'>0$ such that\n\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\begin{enumerate}\n \\item $y \\in B(y',r') \\subset\\subset B(y,r)$,\n\n \\item $x\\in \\operatorname{Leb}{\\rho}$ and $\\rho(x) <+\\infty$,\n\n \\item $x\\in \\operatorname{Leb}{\\rho'}$ and $\\rho'(x)>0$,\n\\end{enumerate}\nwhere $\\rho$ denotes the density of $(\\pi_1)_\\sharp\\gamma$ and $\\rho'$ the density of $(\\pi_1)_\\sharp\\gamma\\lfloor(\\mathbb{H}^n\\times B(y',r'))$ with respect to $\\mathcal{L}^{2n+1}$.\n\\end{lem}\n\n\\begin{proof} Let $(y_m)_{m\\geq 1}$ be a dense sequence in $\\mathbb{H}^n$.\nFor each $m,k \\in \\mathbb{N}^*$, set $\\gamma_{m,k}:=\\gamma \\lfloor (\\mathbb{H}^n \\times B(y_m, r_k))$ where $r_k:=1\/k$. Let $\\rho_{m,k}$ denote the density of $(\\pi_1)_\\sharp\\gamma_{m,k}$ with respect to $\\mathcal{L}^{2n+1}$. Set $A_{m,k} := \\mathbb{H}^n \\setminus (\\operatorname{Leb} \\rho \\cap \\operatorname{Leb} \\rho_{m,k} \\cap \\{\\rho < +\\infty\\})$. We have $\\mathcal{L}^{2n+1}(A_{m,k})=0$. Since $(\\pi_1)_\\sharp\\gamma \\ll \\mathcal{L}^{2n+1}$, it follows that $\\gamma(A_{m,k}\\times B(y_m, r_k)) \\leq (\\pi_1)_\\sharp\\gamma (A_{m,k}) =0$. Next\n \\begin{equation*}\n  \\gamma(\\{\\rho_{m,k}=0\\} \\times B(y_m,r_k)) = (\\pi_1)_\\sharp\\gamma_{m,k}(\\{\\rho_{m,k}=0\\}) = 0.\n \\end{equation*}\nIt follows that $\\gamma(D_{m,k})=0$ for all $m,k \\in \\mathbb{N}^*$ where\n\\begin{equation*}\n D_{m,k}:=\\left[\\mathbb{H}^n \\setminus (\\operatorname{Leb} \\rho \\cap \\operatorname{Leb} \\rho_{m,k} \\cap \\{\\rho < +\\infty\\} \\cap \\{\\rho_{m,k}>0\\}) \\right] \\times B(y_m,r_k)\n\\end{equation*}\nhence $\\gamma(\\cup_{m,k} D_{m,k})=0$ and $\\gamma$ is concentrated on $\\mathbb{H}^n \\setminus \\cup_{m,k} D_{m,k}$. Then the conclusion follows noting that for each $(x,y) \\in \\mathbb{H}^n\\times\\mathbb{H}^n$ and $r>0$, one can find $m,k\\in \\mathbb{N}^*$ such that $y \\in B(y_m,r_k)\\subset \\subset B(y,r)$.\n\\end{proof}\n\nWe say that $x\\in E$ is a Lebesgue point of a Borel set $E$ if $x\\in \\operatorname{Leb} {\\rm 1\\mskip-4mu l}_E$, i.e., if $x\\in E$ and \n\\begin{equation*}\n \\lim_{r\\rightarrow 0} \\dfrac{\\mathcal{L}^{2n+1}(E\\cap B(x,r))}{\\mathcal{L}^{2n+1}(B(x,r))} = 1,\n\\end{equation*}\nand we denote by $\\operatorname{Leb} E:= \\operatorname{Leb} {\\rm 1\\mskip-4mu l}_E$ the set of all Lebesgue points of $E$. Note that $\\mathcal{L}^{2n+1}(E\\setminus \\operatorname{Leb} E) = 0$.\n\nThe next lemma together with Lemma~\\ref{mainlemma} and Lemma~\\ref{pi2} is one of the key ingredients of the proof of Theorem~\\ref{mainthmbis} and eventually of the existence of a solution to Monge's transport problem. It can be recovered as a consequence of Lemma~\\ref{dens1}. However, for sake of clarity, we state and prove it independently.\n\n\\begin{lem}  \\label{dens2}\n Let $\\gamma\\in \\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n)$ be such that $(\\pi_1)_\\sharp\\gamma \\ll \\mathcal{L}^{2n+1}$. Assume that $\\gamma$ is concentrated on a $\\sigma$-compact set $\\Gamma$. For $y\\in\\mathbb{H}^n$ and $r>0$, set \n\\begin{equation*}\n \\Gamma^{-1} (B(y,r)) = \\pi_1(\\Gamma \\cap (\\mathbb{H}^n \\times B(y,r))).\n\\end{equation*}\nThen $\\Gamma^{-1} (B(y,r))$ is a Borel set and $\\gamma$ is concentrated on a set $\\Gamma'\\subset\\Gamma$ such that for all $(x,y)\\in\\Gamma'$ and all $r>0$, $x\\in \\operatorname{Leb}{\\Gamma^{-1} (B(y,r)})$.\n\\end{lem}\n\n\\begin{proof}\n Since $\\Gamma$ is  $\\sigma$-compact, $\\Gamma^{-1} (B(y,r))$ is also $\\sigma$-compact hence a Borel set. Set $A := \\{(x,y) \\in \\Gamma;\\,x\\notin \\operatorname{Leb}\\Gamma^{-1}(B(y,r)) \\text{ for some } r>0 \\}$\nand let us show that $\\gamma(A)=0$. For each $k\\in\\mathbb{N}^*$, consider a countable covering of \n$\\mathbb{H}^n$ by balls $(B(y_i^k,r_k))_{\\,i \\geq 1}$ of radius $r_k:=1\/(2k)$. If $(x,y) \\in \\Gamma$ and $x \\notin \\operatorname{Leb}\\Gamma^{-1}(B(y,r))$ then for any $k \\geq 1\/r$ and $y_i^k$ such that\n$d(y_i^k,y)<r_k$, one has $x\\in \\Gamma^{-1}(B(y_i^k,r_k)) \\setminus \\operatorname{Leb}\\Gamma^{-1}(B(y_i^k,r_k))$. It follows that \n\\begin{equation*}\n\\pi_1(A) \\,\\, \\subset \\,\\, \\bigcup_{k\\geq 1} \\, \\bigcup_{\\,i \\geq 1} \n\\Gamma^{-1}(B(y_i^k,r_k)) \\setminus \\operatorname{Leb}\\Gamma^{-1}(B(y_i^k,r_k)) .\n\\end{equation*}\nThe set on the right-hand side has $\\mathcal{L}^{2n+1}$-measure 0. Since $(\\pi_1)_\\sharp\\gamma \\ll \\mathcal{L}^{2n+1}$, it follows that $\\gamma(A) \\leq  (\\pi_1)_\\sharp\\gamma(\\pi_1(A))=0$.\n\\end{proof}\n\n\n\\section{Lower density of the transport set} \\label{main}\n\nWe consider optimal transport plans in $\\Pi_1(\\mu,\\nu)$ that are obtained as weak limit of solutions to the variational approximations introduced in Section~\\ref{varapprox}. We prove that if $\\gamma$ is such a transport plan then it is concentrated on a set $\\Gamma$ whose related transport set has positive lower density at each point $x\\in \\pi_1(\\Gamma)$ for some suitable notion of lower density. As already mentioned, this is one of the main ingredient in the proof of Theorem~\\ref{mainthmbis}. Following the notion of transport set introduced in e.g.~\\cite{ap}, we define in our geometrical context the transport set related to a set $\\Gamma\\subset \\mathbb{H}^n\\times\\mathbb{H}^n$ as \n\\begin{equation*}\n T(\\Gamma) := \\{ (e_t \\circ S)(x,y);\\, (x,y)\\in \\Gamma, \\, t\\in (0,1) \\}.\n\\end{equation*}\nRecall that $S$ is a measurable selection of minimal curves and that $(e_t \\circ S)(x,y)$ denotes the point at distance $t \\,d(x,y)$ from $x$ on the selected minimal curve $S(x,y)$ between $x$ and $y$, see Subsection~\\ref{sect-interpolation}.\n\n\n\\begin{lem}  \\label{mainlemma}\n Let $\\gamma\\in\\Pi(\\mu,\\nu)$ obtained as a weak limit of solutions to $(P_{\\varepsilon_k})$ for some sequence $\\varepsilon_k$ converging to 0. Then $\\gamma$ is concentrated on a set $\\Gamma$ such that for all $(x,y)\\in\\Gamma$ such that $x\\not=y$ and all $r>0$, we have\n\\begin{equation*}\n \\liminf_{\\delta\\downarrow 0} \\, \\dfrac{\\mathcal{L}^{2n+1}(T(\\Gamma \\cap [B(x,\\frac{\\delta}{2}) \\times B(y,r)]) \\cap B(x,\\delta))}{\\mathcal{L}^{2n+1}(B(x,\\delta))}\\, > 0.\n\\end{equation*}\n\\end{lem}\n\nThe proof below follows the line of the proof of the similar property in~\\cite{champion-dePascale}. In our context it requires however some technical refinement. \n\n \\begin{proof}  We consider the set $\\Gamma$ obtained by Lemma~\\ref{dens1}, $(x,y) \\in \\Gamma$ with $x \\neq y$ and $r>0$. Then let $y'\\in\\mathbb{H}^n$ and $r'>0$ be given by Lemma~\\ref{dens1} so that Lemma~\\ref{dens1}(i), (ii) and (iii) hold. Using the same notations as in this lemma, we set \\begin{equation*}\n G:=\\{z \\in \\mathbb{H}^n;\\, \\ \\frac{1}{2}\\rho'(x) \\leq \\rho' (z) \\,\\, \\mbox{and} \\,\\,\n\\rho(z) \\leq  2\\rho(x) \\}.\n\\end{equation*}\nThen $G$ is a Borel set. We have $0< \\rho'(x) \\leq \\rho(x)$ (remember the convention about densities of absolutely continuous measure, see~\\eqref{e:density}). Since $x\\in \\operatorname{Leb} \\rho \\cap \\operatorname{Leb} \\rho'$, see Lemma~\\ref{dens1}(ii) and (iii), it follows that $x\\in\\operatorname{Leb} G$.\n\nFix $\\delta>0$ such that $\\delta < d(x,y)+r$ and \n\\begin{equation} \\label{e:reg-delta}\n \\frac{1}{2}\\mathcal{L}^{2n+1}(B(x,s)) \\leq \\mathcal{L}^{2n+1}(G \\cap B(x,s))\n\\end{equation}\nfor all $s \\in (0,\\delta)$ and fix $t>0$ such that $4t(d(x,y)+r)< \\delta$.\n\nWe set $G_\\delta:=G\\cap B(x,\\frac{\\delta}{2})$, $A_\\delta:=G_\\delta \\times B(y',r')$ and \n$\\gamma_{\\delta}:= \\gamma \\lfloor A_\\delta$.\nWe shall prove that\n\\begin{equation} \\label{e:main1}\n \\frac{\\rho'(x)}{4} \\, \\mathcal{L}^{2n+1}(B(x,\\frac{\\delta}{2})) \\leq (e_t \\circ S)_\\sharp \\gamma_{\\delta} (B(x,\\delta)) \n\\end{equation} \nand\n\\begin{equation} \\label{e:main2}\n (e_t \\circ S)_\\sharp \\gamma_{\\delta} (B(x,\\delta)) \\leq 2^{2n+4} \\rho(x) \\mathcal{L}^{2n+1} (T(\\Gamma \\cap [ B(x,\\frac{\\delta}{2}) \\times B(y,r)]) \\cap B(x,\\delta)).\n\\end{equation}\nThen \\eqref{e:main1} and \\eqref{e:main2} will yield \n\\begin{equation*}\n2^{-(2n+6)} \\frac{\\rho' (x)}{\\rho(x)}\\, \\mathcal{L}^{2n+1}(B(x,\\frac{\\delta}{2})) \\leq \\mathcal{L}^{2n+1}(T(\\Gamma \\cap [ B(x,\\frac{\\delta}{2}) \\times B(y,r)]) \\cap B(x,\\delta))\n\\end{equation*}\nfor any $\\delta>0$ small enough which completes the proof.\n\nTo prove \\eqref{e:main1}, we note that $(\\pi_1)_\\sharp \\gamma_{\\delta} \\ll \\mathcal{L}^{2n+1}$ with density bounded below by $\\frac{1}{2} \\rho'(x) $ $\\mathcal{L}^{2n+1}$-a.e. on $G_\\delta$. Together with \\eqref{e:reg-delta}, it follows that\n\\begin{equation*}\n \\frac{\\rho'(x)}{4} \\, \\mathcal{L}^{2n+1}(B(x,\\frac{\\delta}{2})) \\leq (\\pi_1)_\\sharp \\gamma_{\\delta} (B(x,\\frac{\\delta}{2})).\n\\end{equation*}\nNext, by choice of $\\delta$ and $t$, we have $(e_t \\circ S)(z,w) \\in B(x,\\delta)$ for all $z \\in B(x,\\frac{\\delta}{2})$ and $w \\in B(y,r)$, hence\n\\begin{equation*}\n B(x,\\frac{\\delta}{2}) \\times B(y',r') \\subset B(x,\\frac{\\delta}{2}) \\times B(y,r) \\subset (e_t \\circ S)^{-1} (B(x,\\delta))\n\\end{equation*}\nand it follows that \n\\begin{equation*}\n (\\pi_1)_\\sharp \\gamma_{\\delta} (B(x,\\frac{\\delta}{2})) = \\gamma_{\\delta} (B(x,\\frac{\\delta}{2}) \\times B(y',r')) \\leq (e_t \\circ S)_\\sharp \\gamma_{\\delta} (B(x,\\delta))\n\\end{equation*}\nand this completes the proof of \\eqref{e:main1}.\n\nWe prove now \\eqref{e:main2}. By hypothesis, $\\gamma$ is a weak limit of solutions $\\gamma_k$ to $(P_{\\varepsilon_k})$ for some sequence $\\varepsilon_k$ converging to 0. For each fixed $k\\in\\mathbb{N}$, we apply Lemma~\\ref{restrictions} with $U = G_\\delta \\times \\mathbb{H}^n$ and Proposition~\\ref{e:Linftydensityestimates} with $\\overline \\mu = (\\pi_1)_\\sharp (\\gamma_k \\lfloor U)$ and $\\overline \\nu = (\\pi_2)_\\sharp (\\gamma_k \\lfloor U)$. Taking into account the fact that $(\\pi_1)_\\sharp (\\gamma_k \\lfloor U) = \\mu \\lfloor G_\\delta$, we get that $(e_t \\circ S)_\\sharp (\\gamma_k \\lfloor G_\\delta \\times \\mathbb{H}^n) \\ll \\mathcal{L}^{2n+1}$ with density in $L^\\infty$ and whose $L^\\infty$-norm is bounded by \n\\begin{equation} \\label{e:rhok}\n \\frac{1}{(1-t)^{2n+3}} \\, \\|\\rho \\lfloor_{G_\\delta} \\|_{L^\\infty} \\leq 2^{2n+4} \\rho(x).\n\\end{equation}\nNext we check that $(e_t \\circ S)_\\sharp (\\gamma_k \\lfloor G_\\delta \\times \\mathbb{H}^n)$ converges weakly to $(e_t \\circ S)_\\sharp (\\gamma \\lfloor G_\\delta \\times \\mathbb{H}^n)$. First it follows from Lemma \\ref{weakres} (to be proved below) that $\\gamma_k\\lfloor G_\\delta \\times \\mathbb{H}^n$ converges weakly to $\\gamma \\lfloor G_\\delta \\times \\mathbb{H}^n$. Then, noting that $\\gamma$ and each $\\gamma_k$ are concentrated on $\\Omega$ and that $e_t \\circ S$ is continuous on $\\Omega$, the claim follows from Lemma \\ref{conv} (to be proved below) applied with $\\overline\\gamma = \\gamma \\lfloor G_\\delta \\times \\mathbb{H}^n$, $\\overline\\gamma_k = \\gamma_k \\lfloor G_\\delta \\times \\mathbb{H}^n$, $B=\\Omega$ and $f = \\varphi \\circ e_t \\circ S$ where $\\varphi \\in C_b(\\mathbb{H}^n)$. The fact that $\\gamma$ is concentrated on $\\Omega$ follows from Lemma \\ref{pi1.1}. To check that $\\gamma_k$ is concentrated on $\\Omega$, denote by $\\{y_i^k\\}_i$ the finite set of the atoms of $(\\pi_2)_\\sharp \\gamma_k$. We have that $\\gamma_k$ is concentrated on $\\mathbb{H}^n\\times \\{y_i^k\\}_i$. On the other hand  $\\gamma_k(L_{ y_i^k} \\times \\{y_i^k\\}) \\leq \\gamma_k(L_{ y_i^k} \\times \\mathbb{H}^n) = \\mu(L_{ y_i^k}) = 0$ since $\\mu \\ll \\mathcal{L}^{2n+1}$. It follows that $\\gamma_k$ is concentrated on $\\cup_i [(\\mathbb{H}^n \\setminus L_{ y_i^k}) \\times \\{y_i^k\\}] \\subset \\Omega$. Then, taking into account \\eqref{e:rhok}, we get\n\\begin{equation*}\n |\\int_{\\mathbb{H}^n} \\varphi \\,\\, d(e_t \\circ S)_\\sharp (\\gamma \\lfloor G_\\delta \\times \\mathbb{H}^n)| \\leq 2^{2n+4} \\rho(x) \\, \\|\\varphi\\|_{L^1}\n\\end{equation*}\nfor every $\\varphi \\in C_b(\\mathbb{H}^n)$. It follows that $(e_t \\circ S)_\\sharp (\\gamma \\lfloor G_\\delta \\times \\mathbb{H}^n)$ is in $(L^1)'$ with density in $L^\\infty$ and whose $L^\\infty$-norm is bounded by $2^{2n+4} \\rho(x)$. Since $(e_t \\circ S)_\\sharp \\gamma_\\delta \\leq (e_t \\circ S)_\\sharp (\\gamma \\lfloor G_\\delta \\times \\mathbb{H}^n)$, the same holds true for $(e_t \\circ S)_\\sharp \\gamma_\\delta$. Finally we note that $\\gamma_\\delta$ being concentrated on $\\Gamma \\cap [ B(x,\\frac{\\delta}{2}) \\times B(y',r')] \\subset \\Gamma \\cap [ B(x,\\frac{\\delta}{2}) \\times B(y,r)]$, the measure $(e_t \\circ S)_\\sharp \\gamma_\\delta$ is concentrated on $T(\\Gamma \\cap [ B(x,\\frac{\\delta}{2}) \\times B(y',r')]) \\subset T(\\Gamma \\cap [ B(x,\\frac{\\delta}{2}) \\times B(y,r)])$. All together we get\n\\begin{equation*}\n\\begin{split}\n (e_t \\circ S)_\\sharp \\gamma_{\\delta} (B(x,\\delta)) &= (e_t \\circ S)_\\sharp \\gamma_{\\delta} (T(\\Gamma \\cap [ B(x,\\frac{\\delta}{2})\n\\times B(y,r)]) \\cap B(x,\\delta) )\\\\\n&\\leq  2^{2n+4} \\rho(x) \\mathcal{L}^{2n+1} (T(\\Gamma \\cap [ B(x,\\frac{\\delta}{2}) \\times B(y,r)]) \\cap B(x,\\delta))\n\\end{split}\n\\end{equation*}\nwhich proves \\eqref{e:main2}.\n\\end{proof}\n\n\\begin{lem} \\label{weakres} \nLet $X$ be a separable and locally compact Hausdorff metric space in which every open set is $\\sigma$-compact. Let $(\\gamma_k)_k$ be a sequence in $\\mathcal P (X\\times X)$\nwhich converges weakly to some $\\gamma \\in \\mathcal P (X\\times X)$ and\nsuch that $(\\pi_1)_\\sharp \\gamma_k = (\\pi_1)_\\sharp \\gamma$ for every $k\\in \\mathbb{N}$. Then for any Borel set $G \\subset X$, the sequence \n$(\\gamma_k \\lfloor  G\\times X)_k$ converges weakly to $\\gamma \\lfloor G\\times X$.\n\\end{lem}\n\n\\begin{proof} We have to prove that for any $\\varphi \\in C_b(X)$,\n\\begin{equation*}\n\\lim_{k\\rightarrow +\\infty} \\int_{X\\times X} {\\rm 1\\mskip-4mu l}_G(x) \\varphi (x,y) \\,d \\gamma_k(x,y) = \\int_{X \\times X} {\\rm 1\\mskip-4mu l}_G(x) \\varphi(x,y) \\,d \\gamma(x,y).\n\\end{equation*}\n\nIt follows from Lusin's Theorem that for any \n$\\varepsilon>0$ there exists a closed set $F_\\varepsilon$ such that ${{\\rm 1\\mskip-4mu l}_G} \\lfloor_{F_\\varepsilon}$ is continuous and $(\\pi_1)_\\sharp \\gamma(X \\setminus\nF_\\varepsilon) < \\varepsilon$. As a consequence, for every $\\varepsilon >0$, the restriction of $(x,y) \\mapsto {\\rm 1\\mskip-4mu l}_G(x) \\varphi (x,y)$ to $F_\\varepsilon \\times X$ is continuous and \n$$\\limsup_{k\\rightarrow +\\infty} \\gamma_k ((X\\setminus F_\\varepsilon) \\times X)\n= (\\pi_1)_\\sharp \\gamma (X \\setminus F_\\varepsilon) < \\varepsilon.$$\nThen since $(x,y)\\mapsto |{\\rm 1\\mskip-4mu l}_G(x) \\varphi (x,y)|$ is\nbounded and hence uniformly integrable with respect to\n$(\\gamma_k)_k$, the claim follows from \\cite[Proposition 5.1.10]{ags}.\n\\end{proof}\n\n\\begin{lem} \\label{conv}\nLet $X$ be a separable metric space and $(\\overline\\gamma_k)_k$ be a sequence in $\\mathcal P(X)$ which converges weakly \nto some $\\overline\\gamma \\in\\mathcal P(X)$. Let $f:X\\to \\mathbb{R}$ be a measurable and bounded function which is continuous \nin $B$ for some Borel set $B\\subset X$ such that $\\overline\\gamma_k(X\\setminus B)=0$ for every $k\\in \\mathbb{N}$ and $\\overline\\gamma(X\\setminus B)=0$, then\n\\begin{equation*}\n \\lim_{k \\to \\infty} \\int_X f d\\gamma_k= \\int_X f d\\gamma.\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof} Let $\\overline{f}$ and $\\tilde{f}$ be respectively the lower and\n upper semicontinuous envelope of $f$. We have $\\overline{f}=f=\\tilde{f}$ on $B$ and hence $\\gamma$-a.e. and $\\gamma_k$-a.e. for every $k\\in\\mathbb{N}$. It follows that\n\\begin{multline*}\n \\int_X f \\, d\\gamma = \\int_X \\overline f \\, d\\gamma \\leq \\liminf_{k \\to \\infty} \\int_X \\overline{f} \\,d\\gamma_k \n= \\liminf_{k \\to \\infty} \\int_X f  \\,d\\gamma_k \\\\\n\\leq \\limsup_{k \\to \\infty} \\int_X f d\\gamma_k \n= \\limsup_{k \\to \\infty} \\int_X \\tilde f d\\gamma_k \\leq \\int_X \\tilde f d\\gamma = \\int_X f \\, d\\gamma\n\\end{multline*}\nwhich proves the claim.\n\\end{proof}\n\n\\section{Solution to Monge's problem} \\label{conclusion}\n\nWe prove that optimal transport plans in $\\Pi_1(\\mu,\\nu)$ that are obtained as weak limit of solutions of the variational approximations introduced in Section~\\ref{varapprox} are induced by a transport, hence giving a solution to Monge's transport problem as stated in Theorem~\\ref{mainthm}. Note that due to the fact that $\\Pi$ is relatively compact in $\\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n)$, such optimal transport plans do exist.\n\n\\begin{thm} \\label{mainthmbis}\n Let $\\varepsilon_k$ be a sequence converging to 0 and $\\gamma_{\\varepsilon_k}$ a sequence of solutions to $(P_{\\varepsilon_k})$ which is weakly converging to some $\\gamma\\in \\mathcal P(\\mathbb{H}^n\\times\\mathbb{H}^n)$. Then $\\gamma$ is concentrated on a $\\mu$-measurable graph and hence induced by a transport.\n\\end{thm}\n\n\\begin{proof}\nFirst we know from Lemma~\\ref{optpi2} that $\\gamma\\in \\Pi_2(\\mu,\\nu)$. From the previous sections and using inner regularity of Borel probability measures, one can then find $\\sigma$-compact sets $\\Gamma$ and $\\Gamma'$ such that $\\Gamma'\\subset \\Gamma \\subset \\Omega$ and the conclusions of Lemma~\\ref{pi2}, Lemma~\\ref{dens2} and Lemma~\\ref{mainlemma} hold. We prove here that for any $x\\in \\pi_1(\\Gamma')$ there is a unique $y\\in \\mathbb{H}^n$ such that $(x,y)\\in \\Gamma'$.\n\nBy contradiction, assume that one can find $x_0 \\in \\pi_1(\\Gamma')$ and $(x_0,y_0)\\in \\Gamma$, $(x_0,y_1)\\in \\Gamma$ with $y_0\\not=y_1$. Without loss of generality one can assume that $d(x_0,y_0) \\leq d(x_0,y_1)$ and $x_0\\not=y_1$. Then, by Lemma \\ref{dens2} and Lemma \\ref{mainlemma}, for all $r>0$ and for all $\\delta>0$ small enough, one can find $x'\\in B(x_0,\\delta) \\cap \\Gamma^{-1} (B(y_0,r)) \\cap T(\\Gamma \\cap [B(x_0,\\frac{\\delta}{2}) \\times B(y_1,r)])$. It follows that one can find $y' \\in B(y_0,r)$ such that $(x',y') \\in \\Gamma$ and $(x,y) \\in \\Gamma \\cap (B(x_0,\\frac{\\delta}{2})\\times B(y_1,r))$ such that $x\\not=y$, $x'\\not=x$ and $x'$ lie on the minimal curve between $x$ and $y$. Then it follows from Lemma~\\ref{pi2} that $x$, $x'$, $y$ and $y'$ lie on the same minimal curve ordered in that way.\n\nAssume first that $ d(x_0,y_0) < d(x_0,y_1)$. We know from Lemma \\ref{pi2} that $d(x,y)\\leq d(x,y')$. On the other hand, we have\n\\begin{equation*}\n\\begin{split}\n d(x,y')&\\leq d(x,x_0) + d(x_0,y_0) + d(y_0,y')\\\\\n& \\leq d(x_0,y_0) + \\dfrac{\\delta}{2} + r \\\\\n&= d(x_0,y_1) + d(x_0,y_0) - d(x_0,y_1) + \\dfrac{\\delta}{2} + r \\\\\n&\\leq  d(x_0,x) + d(x,y) + d(y,y_1) + d(x_0,y_0) - d(x_0,y_1) + \\dfrac{\\delta}{2} + r\\\\\n&\\leq d(x,y) + d(x_0,y_0) - d(x_0,y_1) + \\delta + 2r.\n\\end{split}\n\\end{equation*}\nIt follows that $d(x,y')< d(x,y)$ provided we take $r>0$ and $\\delta>0$ small enough which gives a contradiction. If $ d(x_0,y_0) = d(x_0,y_1)$, we have\n\\begin{equation*}\n\\begin{split}\nd(x_0,y_1) &\\leq d(x_0,x) + d(x,y) + d(y,y_1) \\\\\n& = d(x_0,x) + d(x,y') - d(y',y) +  d(y,y_1)\\\\\n& \\leq d(x,y') - d(y',y) + \\dfrac{\\delta}{2} + r ,\n\\end{split}\n\\end{equation*}\n\\begin{equation*}\n d(x,y') \\leq d(x,x_0) + d(x_0,y_0) + d(y_0,y') \\leq d(x_0,y_0) + \\dfrac{\\delta}{2} + r,\n\\end{equation*}\n\\begin{equation*}\n d(y',y) \\geq d(y_0,y_1) - d(y_0,y') - d(y_1,y)\\geq d(y_0,y_1) -2r,\n\\end{equation*}\nhence,\n\\begin{equation*}\n d(x_0,y_1) \\leq d(x_0,y_0) - d(y_0,y_1) + 4r +\\delta. \n\\end{equation*}\nIt follows that $d(x_0,y_1) < d(x_0,y_0)$ provided we take $r>0$ and $\\delta>0$ small enough which gives also a contradiction.\n\\end{proof}\n\n\n\\section{Extension to more general metric measure spaces} \\label{extensions}\n\nFirst we note that a major part of intermediate steps in the strategy adopted in the present paper can be naturally extended to Polish and non-branching geodesic spaces equipped with a reference measure for which the Lebesgue's differentiation theorem holds. \n\nNext our choice of approximating costs $c_\\varepsilon$ in the approximation procedure is not the only possible one. This choice could in particular be adapted to fit other contexts (for instance concerning the relevant properties of solutions to the transport problem associated to the approximating cost).\n\nFinally the Measure Contraction Property is here technically very convenient. We note however that this property is unnecessarily too strong for what is actually needed in the proof about the lower density of the transport set. Much local and weaker versions about the behavior of the measure of sets transported along minimal curves are indeed sufficient as clearly shows up from the proof.\n\nThis approach can in particular be adapted to give an alternative proof of the existence of solutions to Monge's transport problem in the Riemannian setting without using Sudakov's type arguments.\n\nFor the reasons listed above it is furthermore very likely that the present strategy could be adapted and extended to other geodesic metric spaces.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\@ifstar{\\origsection*}{\\mysection}{\\@ifstar{\\origsection*}{\\mysection}}\n\\def\\mysection{\\@startsection{section}{1}\\z@{.7\\linespacing\\@plus\\linespacing}{\n.5\\linespacing}{\\normalfont\\scshape\\centering\\S}}\n\\makeatother\n\n\\usepackage{comment}\n\\usepackage{amsmath,amssymb,amsthm}\n\\usepackage{mathrsfs}\n\\usepackage{dsfont}\n\\usepackage{mathabx}\\changenotsign\n\\usepackage[ruled,linesnumbered,vlined]{algorithm2e}\n\n\\usepackage{tikz}\n\\usetikzlibrary{shapes,decorations,arrows,calc,arrows.meta,fit,positioning}\n\\usepackage{graphicx}\n\\usepackage{fullpage}\n\\usetikzlibrary{decorations.pathreplacing, matrix}\n\n\\usepackage{lineno}\n\n\\usepackage{microtype}\n\\usepackage{color}\n\\usepackage[utf8]{inputenc}\n\\usepackage{enumitem}\n\\newcommand\\rmlabel{\\upshape({\\itshape\\roman*\\,\\\/})}\n\\newcommand\\RMlabel{\\upshape(\\Roman*)}\n\\newcommand\\alabel{\\upshape({\\itshape\\alph*\\,\\\/})}\n\\newcommand\\Alabel{\\upshape({\\itshape\\Alph*\\,\\\/})}\n\\newcommand\\nlabel{\\upshape({\\itshape\\arabic*\\,\\\/})}\n\\newcommand\\nlabelp{\\upshape(P\\arabic*)}\n\n\\usepackage{xcolor}\n\\usepackage{hyperref}\n\\hypersetup{%\n    colorlinks,\n    linkcolor={red!60!black},\n    citecolor={green!60!black},\n    urlcolor={blue!60!black}\n}\n\n\\usepackage{tikz}\n\\usepackage{subcaption}\n\\usetikzlibrary{graphs}\n\n\\usepackage{bookmark}\n\n\\usepackage[T1]{fontenc}\n\\usepackage{lmodern}\n\n\\usepackage[english]{babel}\n\n\\usepackage{fullpage}\n\\usepackage{setspace}\n\\footskip28pt\n\n\\usepackage{enumitem}\n\n\\def\\itm#1{\\rm ({#1})}\n\\def\\itmit#1{\\itm{\\it #1\\,}}\n\\def\\itmit{\\roman{*}}{\\itmit{\\roman{*}}}\n\n\\newcommand{\\hat{r}}{\\hat{r}}\n\\let\\eps=\\varepsilon\n\\let\\polishlcross=\\ifmmode\\ell\\else\\polishlcross\\fi\n\\def\\ifmmode\\ell\\else\\polishlcross\\fi{\\ifmmode\\ell\\else\\polishlcross\\fi}\n\n\\newcommand\\tand{\\ \\text{and}\\ }\n\\newcommand\\qand{\\quad\\text{and}\\quad}\n\\newcommand\\qqand{\\qquad\\text{and}\\qquad}\n\n\\let\\emptyset=\\varnothing\n\\let\\setminus=\\smallsetminus\n\\let\\backslash=\\smallsetminus\n\n\\newcommand{\\mathbb{E}}{\\mathbb{E}}\n\\newcommand{\\mathbb{N}}{\\mathbb{N}}\n\\newcommand{\\mathbb{P}}{\\mathbb{P}}\n\n\\makeatletter\n\\def\\mathpalette\\mov@rlay{\\mathpalette\\mov@rlay}\n\\def\\mov@rlay#1#2{\\leavevmode\\vtop{%\n   \\baselineskip\\z@skip \\lineskiplimit-\\maxdimen\n   \\ialign{\\hfil$\\m@th#1##$\\hfil\\cr#2\\crcr}}}\n\\newcommand{\\charfusion}[3][\\mathord]{\n    #1{\\ifx#1\\mathop\\vphantom{#2}\\fi\n        \\mathpalette\\mov@rlay{#2\\cr#3}\n      }\n    \\ifx#1\\mathop\\expandafter\\displaylimits\\fi}\n\\makeatother\n\n\\newcommand{\\charfusion[\\mathbin]{\\cup}{\\cdot}}{\\charfusion[\\mathbin]{\\cup}{\\cdot}}\n\\newcommand{\\charfusion[\\mathop]{\\bigcup}{\\cdot}}{\\charfusion[\\mathop]{\\bigcup}{\\cdot}}\n\\newcommand{\\abs}[1]{\\left| #1 \\right|}\n\\newcommand{\\cbc}[1]{\\left\\lbrace #1 \\right\\rbrace}\n\\newcommand{\\bc}[1]{\\left( #1 \\right)}\n\\newcommand{\\floor}[1]{\\left\\lfloor #1 \\right\\rfloor}\n\\newcommand{\\ceil}[1]{\\left\\lceil #1 \\right\\rceil}\n\\newcommand{\\mhk}[1]{\\textcolor{red}{#1}}\n\\newcommand{\\op}[1]{\\textcolor{blue}{#1}}\n\n\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{conjecture}[theorem]{Conjecture}\n\\newtheorem{problem}[theorem]{Problem}\n\\newtheorem{question}[theorem]{Question}\n\\newtheorem{property}[theorem]{Property}\n\\newtheorem{claim}[theorem]{Claim}\n\\newtheorem{subclaim}[theorem]{Sub-Claim}\n\\newtheorem{proposition}[theorem]{Proposition}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{fact}[theorem]{Fact}\n\\newtheorem*{observation}{Observation}\n\n\\def\\mcarrow{\\xrightarrow[{\\raisebox{.5mm}[1mm][0mm]{$\\scriptstyle \\rm\np$}}]{\\raisebox{0.0mm}[0mm]{$\\scriptstyle \\rm rb$}}}\n\n\n\\def\\pmc#1{p^{\\rm rb}_{#1}}\n\\def\\mathcal{P}{\\mathcal{P}}\n\\def\\mathcal{W}{\\mathcal{W}}\n\\def\\mathcal{Z}{\\mathcal{Z}}\n\\def\\mathcal{A}{\\mathcal{A}}\n\\def\\mathcal{B}{\\mathcal{B}}\n\n\\usepackage{datetime}\n\\usepackage{lineno}\n\\newcommand*\\patchAmsMathEnvironmentForLineno[1]{%\n\\expandafter\\let\\csname old#1\\expandafter\\endcsname\\csname #1\\endcsname\n\\expandafter\\let\\csname oldend#1\\expandafter\\endcsname\\csname end#1\\endcsname\n\\renewenvironment{#1}%\n{\\linenomath\\csname old#1\\endcsname}%\n{\\csname oldend#1\\endcsname\\endlinenomath}\n\\newcommand*\\patchBothAmsMathEnvironmentsForLineno[1]{%\n\\patchAmsMathEnvironmentForLineno{#1}%\n\\patchAmsMathEnvironmentForLineno{#1*}}%\n\\AtBeginDocument{%\n\\patchBothAmsMathEnvironmentsForLineno{equation}%\n\\patchBothAmsMathEnvironmentsForLineno{align}%\n\\patchBothAmsMathEnvironmentsForLineno{flalign}%\n\\patchBothAmsMathEnvironmentsForLineno{alignat}%\n\\patchBothAmsMathEnvironmentsForLineno{gather}%\n\\patchBothAmsMathEnvironmentsForLineno{multline}%\n}\n\n\\newcommand{}{}\n\\def\\hfill\\scalebox{.6}{$\\Box$}{\\hfill\\scalebox{.6}{$\\Box$}}\n\n\\newenvironment{claimproof}[1][Proof]{\n  \\renewcommand{}{\\qedsymbol}\n  \\renewcommand{\\qedsymbol}{\\hfill\\scalebox{.6}{$\\Box$}}\n  \\begin{proof}[#1]\n}{\n  \\end{proof}\n  \\renewcommand{\\qedsymbol}{}\n}\n\n\\newcommand{\\note}[1]{\\textcolor{red}{#1}}\n\n\\begin{document}\n\\onehalfspace\n\\shortdate\n\\yyyymmdddate\n\\settimeformat{ampmtime}\n\\date{\\today, \\currenttime}\n\n\\title{Minimum degree conditions for containing an $r$-regular $r$-connected subgraph}\n\n\\author[M.~Hahn-Klimroth]{Max Hahn-Klimroth} \n\\address{hahnklim@math.uni-frankfurt.de, Goethe University Frankfurt, Robert-Mayer-Str. 10, 60235 Frankfurt, Germany }\n\\thanks{MHK is supported by DFG grant CO 646\/5.}\n\n\\author[O.~Parczyk]{Olaf Parczyk}\n\\address{parczyk@mi.fu-berlin.de, FU Berlin, Arnimallee 3, 14195 Berlin, Germany}\n\\thanks{OP is supported by DFG grant PA 3513\/1-1.}\n\n\n\\author[Y.~Person]{Yury Person}\n\\address{yury.person@tu-ilmenau.de, TU Ilmenau, Weimarer Str. 25, 98684 Ilmenau, Germany} \n\\thanks{YP is supported by the Carl Zeiss Foundation and by DFG grant PE 2299\/3-1.}\n\n\\begin{abstract}\nWe study  optimal minimum degree conditions when an $n$-vertex graph $G$ contains an $r$-regular $r$-connected subgraph. \nWe prove for $r$ fixed and $n$ large the condition to be $\\delta(G) \\ge \\frac{n+r-2}{2}$ when $nr \\equiv 0 \\pmod 2$. This answers a question of M.~Kriesell.\n\\end{abstract}\n\n\n\\maketitle\n\n\n\n\\@ifstar{\\origsection*}{\\mysection}{Introduction}\n\nA typical question in extremal graph theory is to determine (asymptotically) optimal minimum degree conditions for a graph $G$ on $n$ vertices to contain a given copy of some spanning graph. \nThe classical theorem of Dirac~\\cite{dirac} asserts the optimal minimum degree condition to contain a Hamilton cycle to be $\\tfrac n2$. There are numerous generalisations of this result to higher connected cycles (powers of Hamilton cycles)~\\cite{KSS_Seynmour}, which in turn generalise  the theorems of Corradi and Hajnal~\\cite{CH63} and  Hajnal and Szemer\\'edi~\\cite{HS_erdos} about clique factors in graphs. The most comprehensive result which asymptotically subsumes all of the mentioned results is the bandwidth theorem of B\\\"ottcher, Schacht and Taraz~\\cite{BST09}. This theorem provides a sufficient condition, which asymptotically depends only on the chromatic number of a bounded degree graph with sublinear bandwidth to be contained in a given dense graph. We also refer to the excellent survey~\\cite{KO09} by K\\\"uhn and Osthus for more results.\n\nThe present work is motivated by a question of Matthias Kriesell~\\cite{MKcomm} about optimal minimum degree condition sufficient to assert the existence of a $4$-regular $4$-connected spanning subgraph. This question in turn was motivated by the work of Bang-Jensen and Kriesell on good acyclic orientations of $4$-regular $4$-connected graphs~\\cite{BJK19}.\n\nWe answer Kriesell's question by proving the following general result about $r$-connected $r$-regular subgraphs of $G$.\n\\begin{theorem}\n     \\label{thm:main}\n    For any $r \\ge 2$ there exists an $n_0$ such that any $n$-vertex graph $G$ with minimum degree $\\delta(G) \\ge \\frac{n+r-2}{2}$, $n \\ge n_0$, and $nr \\equiv 0 \\pmod 2$ contains a spanning $r$-regular $r$-connected subgraph.   \n\\end{theorem}\nNote that for $r \\ge 2$ an $n$-vertex graph $G$ with minimum degree $\\delta(G) \\ge \\frac{n+r-2}{2}$ always is $r$-connected, whereas one can easily come up with examples certifying the optimality of this result (e.g.\\ two $K_{(n+r)\/2}$'s sharing $r$ vertices).\nThe theorem above asserts that  there are minimal $r$-connected subgraphs of $G$ which are in fact $r$-regular.\nObserve that for $r=2$ this follows immediately from Dirac's theorem~\\cite{dirac} with $n_0=3$, as  a Hamilton cycle  is $2$-regular and $2$-connected.\nOwing to the use of the regularity lemma the $n_0$ given by Theorem~\\ref{thm:main} will be very large.\n\nIn the following we briefly introduce some notation and discuss possible candidates for $r$-regular $r$-connected subgraphs that will be found  in $G$ by Theorem~\\ref{thm:main}.  \nThe \\emph{$t$-blow-up of a graph $F$} is obtained by replacing every vertex by $t$ vertices and every edge by a complete bipartite graph $K_{t,t}$.\nLet $C_n$ be the cycle on $n$ vertices and $P_n$ the $n$-vertex path.\nWe denote by $C_n(t)$ and $P_n(t)$ the $t$-blow-up of $C_n$ and $P_n$, respectively.\nWe use a similar definition for odd values of $t$.\nWe denote by $C_n(t-\\tfrac12)$ the $t$-blow-up of $C_n$ for $n$ even, where every other edge only gets a $K_{t,t}$ minus a perfect matching.\nSimilarly, $P_n(t-\\tfrac12)$ is the $t$-blow-up of the $n$-vertex path, where every other edge (starting with the first) only gets a $K_{t,t}$ minus a perfect matching.\nWe also call these the \\emph{$(t-\\tfrac 12)$-blow-ups}.\nNote that $C_n(t)$ is $2t$-regular and $C_n(t-\\tfrac 12)$ is $(2t-1)$-regular.\n\nIn most cases in our proof of Theorem~\\ref{thm:main} we will be able to find a spanning copy of an $\\tfrac r2$-blow-up of a cycle, while allowing other structures with all but a small fraction of vertices in $\\tfrac r2$-blow-ups of paths (see Section~\\ref{sec:o_constructions} for more details).\nHowever, when $n$ is even and not divisible by $4$, the graph $G$ obtained by taking the disjoint union of two cliques $K_{n\/2-2}$ and adding four additional vertices that are connected to all previous $n-4$ vertices cannot contain a copy of $C_n(4)$. \nFinally, observe, that the bandwidth theorem~\\cite{BST09} guarantees the asymptotically best minimum degree condition $\\tfrac n2 +o(n)$. Thus, Theorem~\\ref{thm:main} improves this asymptotic bound to the exact one.\n\nBeyond these blow-ups it would be interesting to study the minimum degree threshold for other spanning structures that can be obtained by identifying vertices or edges of copies of a small graph on a cycle.\nIn particular, when the small graph is not bipartite, this threshold can depend on its chromatic number or critical chromatic number similarly as when taking disjoint copies (see~\\cite{KO09}).\n\n\n\\subsection{Organisation of the paper}\nThe paper is structured as follows. In Section~\\ref{sec:tools} we collect the essential tools (regularity and blow-up lemmas), while Section~\\ref{sec:overview} provides a proof overview, which consists of three cases (extremal case I, extremal case II and non-extremal case). These cases are dealt with in the subsequent Sections~\\ref{sec:extremal1},~\\ref{sec:extremal2} and~\\ref{sec:non-extremal}.\n\n\\@ifstar{\\origsection*}{\\mysection}{Tools and Notation}\\label{sec:tools}\nFor standard graph theoretic definitions we refer to Bollob\\'as~\\cite{Bolbook98}.  \nThe main tools are Szemer\u00e9di's regularity lemma~\\cite{Sze_regularity} and the blow-up lemma by Koml\u00f3s, S\u00e1rk\u00f6zy, and Szemer\u00e9di~\\cite{KSS_Blowup}.\nFor this let $G=(V,E)$ be a graph.\nFor any two sets $A,B \\subseteq V$ we denote by $e_G(A,B)$ the number of edges of $G$ with one endpoint in $A$ and one in $B$.\nThen the \\emph{density} $d(A,B)$ between these sets is $\\frac{e(A,B)}{|A||B|}$.\n\n\\begin{definition}\n    The pair $(A,B)$ is \\emph{$\\varepsilon$-regular} if for all $X \\subseteq A$, $Y \\subseteq B$ with $|X| \\ge \\varepsilon |A|$, $|Y| \\ge \\varepsilon |B|$ we have $|d(X,Y)-d(A,B)| \\le \\varepsilon$.    \n\\end{definition}\n\nThe following lemma guarantees that (not too small) induced subgraphs of  $\\varepsilon$-regular pairs are still regular (although with a slightly worse parameter).\n\\begin{lemma}[Slicing lemma]\n\\label{lem:slicing}\nLet $(A,B)$ be an $\\eps$-regular pair with $d(A,B)=d$, let $\\tfrac 12 \\ge \\gamma > \\eps$, and $A' \\subseteq A$ and $B' \\subseteq B$ be of size $|A'| \\ge \\gamma |A|$ and $|B'|\\ge \\gamma |B|$.\nThen $(A',B')$ is $2\\eps$-regular pair with $d(A,B) \\ge d'$, where $|d-d'|\\le \\eps$.\\qed\n\\end{lemma}\n\nWhen working with the regular pairs, one often needs a somewhat stronger concept of super-regularity.\n\\begin{definition}\n    The pair $(A,B)$ is an \\emph{$(\\varepsilon,\\delta)$-super-regular} pair if it is $\\varepsilon$-regular and $\\deg(a,B) \\ge \\delta |B|$, $\\deg(b,A) \\ge \\delta |A|$ for all $a \\in A$, $b \\in B$.\n\\end{definition}\n\nThe next lemma asserts that there every $\\varepsilon$-regular pair contains an almost spanning super-regular pair.\n\\begin{lemma}\n\\label{lem:superreg}\nLet $(A,B)$ be an $\\eps$-regular pair with $d(A,B)=d$.\nThen there exists $A'\\subseteq A$ and $B' \\subseteq B$ with $|A'|\\ge (1-\\eps) |A|$ and $|B'| \\ge (1-\\eps) |B|$ such that $(A',B')$ is a $(2\\eps,d-3\\eps)$-super-regular pair. \\qed\n\\end{lemma}\n\nWe will use the following degree form of the regularity lemma by Koml\u00f3s and Simonovits~\\cite{koml_simon}.\n\n\\begin{lemma}[Regularity lemma, degree version]\n    \\label{lem:regularity}\n    For every $\\varepsilon>0$ there exists an integer $M$ such that for any graph $G$ and $d \\in [0,1]$ there is a partition of $V(G)$ into $\\ell+1 \\le M$ clusters $V_0,\\dots,V_{\\ell}$ and a subgraph $G'$ of $G$ such that\n    \\begin{enumerate}[label=\\upshape(P\\arabic*)]\n        \\item \\label{reg:size} $|V_0| \\le \\varepsilon |V(G)|$ and  $|V_i| = L \\le \\varepsilon |V(G)|$ for all $1 \\le i \\le \\ell$.\n        \\item \\label{reg:deg} $\\deg_{G'}(v) \\ge \\deg_G(v) - (d+\\varepsilon) |V|$ for all $v \\in V$.\n        \\item \\label{reg:ind} For $1 \\le i \\le \\ell$ the set $V_i$ is independent in $G'$.\n        \\item \\label{reg:reg} For $1 \\le i < j \\le \\ell$ the pair $(V_i,V_j)$ is $\\varepsilon$-regular in $G'$ and has density $0$ or $d$.\n    \\end{enumerate}\n\\end{lemma}\n\nThe blow-up lemma allows us to embed spanning subgraphs with bounded degree. We will use the following special case deduced from~\\cite[Remark~13]{KSS_Blowup}.\n\n\\begin{lemma}[Bipartite blow-up lemma]\n    \\label{lem:blowup}\n    For each $d,c>0$ and integer $\\Delta$ there exist $\\varepsilon>0$, $\\alpha>0$ and integer $n_0$ such that the following holds for any $n \\ge n_0$.\n    Let $H$ be a bipartite graphs on classes $A$ and $B$ with $|A|=|B|=n$ such that $(A,B)$ is a $(\\varepsilon,d)$-super-regular pair and let $G$ be a bipartite graph on classes $X$ and $Y$ with $|X|=|Y|=n$ that has maximum degree bounded by $\\Delta$.\n    Moreover, for any $X' \\subseteq X$ and $Y' \\subseteq Y$ with $|X'|,|Y'| \\le \\eps n$ let $A_x \\subseteq A$ and $B_y \\subseteq B$ for each $x \\in X'$ and $y \\in Y'$ with $|A_x|,|B_x| \\ge cn$.\n    Then there exists an embedding of $G$ into $H$ such that all $x \\in X'$ and $y \\in Y'$ are embedded into $A_x$ and $B_y$, respectively.\n\\end{lemma}\n\nWe remark that in our application $X'$ and $Y'$ will be of constant size and all $A_x$ and all $B_y$ will be the same.\n\n\\@ifstar{\\origsection*}{\\mysection}{Proof overview}\\label{sec:overview}\n\nThe proof of Theorem~\\ref{thm:main} will be split into three cases.\nWe now explain this case distinction and then give an overview of the proof for each of these cases.\nLet $G$ be a graph with minimum degree $\\tfrac{n+r-2}{2}$.\nFor $\\alpha>0$ we call $G$ \\emph{$\\alpha$-extremal} if there are two sets $A,B \\subseteq V(G)$ of size $(\\frac{1}{2}-\\alpha)n \\le |A|,|B| \\le \\frac{n}{2}$ such that $d(A,B) < \\alpha$.\nWith the help of the regularity lemma we will cover the case that $G$ is not $\\alpha$-extremal for any $\\tfrac{1}{32}>\\alpha>0$ in Section~\\ref{sec:non-extremal}.\n\nSo we can assume that $G$ is $\\alpha$-extremal for some $\\alpha>0$.\nUsing the minimum degree condition in $G$ it is easy to see that the sets $A$ and $B$ have to be almost disjoint or almost the same.\nThis implies that $G$ contains a large set that is 'almost' independent or it is 'close' to the disjoint union of two cliques $K_{n\/2}$.\nMore precisely, there exists $\\alpha'>0$ such that one of the following holds:\nEither, there are two disjoint sets $A,B \\subseteq V(G)$ with $(\\tfrac 12 -\\alpha')n \\le |A|,|B| \\le (\\tfrac 12 +\\alpha')n$ such that $G[A]$ and $G[B]$ have minimum degree $(\\tfrac 12 -3\\alpha')n$ and every vertex outside of $A \\cup B$ has degree at least $\\alpha'n$ into $A$ and $B$ -- this will be the first extremal case treated in Section~\\ref{sec:extremal1}.\nOr, there is one set $A \\subseteq V(G)$ with $(\\tfrac12 -\\alpha')n \\le |A| \\le (\\tfrac12 +\\alpha')n$ is such that any vertex in $A$ has degree at least $(\\tfrac12 -3\\alpha')n$ into $V(G) \\setminus A$ and every vertex outside of $A$ has degree at least $3 \\alpha' n$ into $A$ -- this is the second extremal case treated in Section~\\ref{sec:extremal2}.\n\nTherefore, when choosing $0<\\alpha< \\tfrac{1}{32}$ sufficiently small for both extremal cases and the remaining cases will be `non-extremal'.\nThis implies Theorem~\\ref{thm:main}.\nIn the remainder of this section we sketch the argument for each of the three cases and afterwards explain why our constructions are indeed $r$-connected.\n\n\\subsection{Non-Extremal Case}\n\\label{sec:o_non-extremal}\n\nWe would like to find a spanning copy of $C_k(\\tfrac r2)$ in $G$, but an obvious necessary condition for this is that $v(G) \\equiv 0 \\pmod{2 \\lceil \\tfrac r2 \\rceil}$.\nIf this condition is satisfied, we will succeed, and, otherwise, find a slightly locally modified version.\nFor the proof we will have constants\n\\begin{align*}\n    \\varepsilon \\ll \\nu \\ll d \\ll \\beta \\ll \\alpha < \\frac{1}{32}\n\\end{align*}\nand $s = \\lceil \\tfrac r2 \\rceil$.\nWe follow similar arguments as in~\\cite{KSS_square}, which can be summarised by the following procedure:\n\\begin{enumerate}[label=\\upshape\\bf Step \\arabic*]\n    \\item Apply regularity lemma (Lemma~\\ref{lem:regularity}) with $\\varepsilon$ and $d$ to obtain a regular partition of $G$.\\label{step:regularise}\n    \\item Find $\\ell$ $\\varepsilon$-regular pairs $(X_i,Y_i)$ covering all but a small set $V_0$ with $|V_0| \\le 20 dn$.\\label{step:matching}\n    \\item For $i=1,\\dots,\\ell$ connect $Y_i$ to $X_{i+1}$ with the $\\tfrac r2$-blow-up of a path that we denote by $P_i$.\\label{step:connect}\n    \\item For $i=1,\\dots,\\ell$ turn $(X_i,Y_i)$ into an $(\\varepsilon,d-\\varepsilon)$-super-regular pair with $|X_i|=|Y_i|$, slightly increasing $V_0$ to $|V_0| \\le 23 dn$.\\label{step:superreg}\n    \\item Repeatedly take $\\nu n$ vertices from $V_0$ and append them to the paths $P_i$.\\label{step:absorb}\n    \\item For $i=1,\\dots,\\ell$ use blow-up lemma (Lemma~\\ref{lem:blowup}) to find a spanning copy of an $\\tfrac r2$-blow-up of a path in $(X_i,Y_i)$ connecting $P_{i-1}$ with $P_{i}$.\\label{step:spanning}\n\\end{enumerate}\n\nThe index $\\ell+1$ corresponds to $1$.\n\\ref{step:regularise} is natural and for~\\ref{step:matching} it is enough to find a large matching in a graph with minimum degree close to $\\frac{n}{2}$.\nDuring the performance of~\\ref{step:absorb} the degree of some vertices might get too small.\nIn this case we add them to a set $Q$ that we take care of before the next round.\nThis terminates as in every execution there are at most $3 \\varepsilon n \\ll \\nu n$ vertices added to $Q$.\nApart from this~\\ref{step:absorb} is very similar to~\\ref{step:connect}, which we now sketch with more details.\n\nLet $X$, $Y$ be the clusters that we want to connect with the $\\tfrac r2$-blow-up of a path $P$.\nIf there is a cluster $Z$ such that $(X,Z)$ and $(Z,Y)$ are $\\varepsilon$-regular pairs with density at least $d$ then we can easily find this path.\nOtherwise, let $A$ be the union of all clusters $Z$ such that $(X,Z)$ is an $\\varepsilon$-regular pair with density at least $d$ and $B$ the union of all clusters $Z$ for $(Y,Z)$ analogously.\nBy the minimum degree property in the cluster graph we get $|A|,|B| \\ge (\\frac{1}{2}-\\alpha)n$.\nAs $G$ is not $\\alpha$-extremal we have $d(A,B) > \\alpha$.\nTherefore, there exist two clusters $Z_1 \\in A$ and $Z_2 \\in B$ with $d(Z_1,Z_2) \\ge \\alpha$ and then $(X,Z_1)$, $(Z_1,Z_2)$, and $(Z_2,Y)$ are $\\varepsilon$-regular pairs with density at least $d$.\nThen it is again easy to find the path that we are interested in by following these three regular pairs.\n\nWe have to ensure that the end vertices of the paths always have high degree into the other cluster of the respective super-regular pair, because we want to connect them later and keep them through~\\ref{step:superreg}.\nFurthermore, we have to ensure that in~\\ref{step:absorb} the sizes of the $(\\varepsilon,d-\\varepsilon)$-super-regular pairs remain balanced.\nWe will give the details in Section~\\ref{sec:non-extremal}.\n\n\\subsection{Extremal Case I}\n\\label{sec:o_extremal1}\n\nIn this extremal case we will not use the regularity lemma, but the blow-up lemma will be helpful.\nRecall that in this case $G$ is `close' to the union of two disjoint cliques of size roughly $\\tfrac n2$ on vertex sets $A$ and $B$.\nThe main challenge is to find a bridge that connects both these cliques.\nIt is then easy to find the desired structure using the high degrees.\n\\begin{enumerate}[label=\\upshape\\bf Step \\arabic*]\n\\item \\label{stepd:bridge} In the case when $r$ is even the bridge will be a matching of size $r$ between $A$ and $B$ such that the end-vertices are well connected on their side.\nThe odd case is a little more delicate and we will find a matching of size $r+1$ or $r$ depending on the size of $V(G) \\setminus \\bc{A \\cup B}$ and the parity of $A$ and $B$.\n\\item \\label{stepd:absorb} Absorb all vertices that do not not belong to $A$ or $B$ by extending both ends of the path.\nWe can ensure that the left-over on each side has size divisible by $2r$.\n\\item \\label{stepd:cover} It is easy to see that the left-over on both sides can be split into a super-regular pair and that we can cover both with the $\\tfrac r2$-blow-up of a path using Lemma~\\ref{lem:blowup}.\n\\end{enumerate}\nIf we take care of the end-tuples between each of the steps this gives an $r$-regular $r$-connected path-structure covering $G$.\nIn Section~\\ref{sec:extremal1} we will give the details of the even and odd case separately.\n\n\\subsection{Extremal Case II}\n\\label{sec:o_extremal2}\n\nAgain, we will not use the regularity lemma in this part, but the blow-up lemma will still be helpful.\nWe can assume that we have a partition of $V(G)$ into $A$ and $B$ of size $(\\tfrac 12 \\pm \\alpha)n$ such that between these sets we have minimum degree $\\alpha n$ and all but at most $\\alpha n$ vertices from $A$ (or $B$) have degree $|B|-\\alpha n$ (or $|A|-\\alpha n$) into $B$ (or $A$).\nW.l.o.g.~assume that $|A|+m=\\tfrac 12 n=|B|-m$, where $0 \\le m \\le \\alpha n$.\nNote that in $G[B]$ we have minimum degree at least $m+\\tfrac{r-2}{2}$.\nLet $s=\\lceil \\tfrac r2 \\rceil$.\n\\begin{enumerate}[label=\\upshape\\bf Step \\arabic*]\n    \\item \\label{stepe:find} If $\\Delta(G[B]) \\le 2 r \\alpha n$ find $m$ copies of $K_{1,s}$, such that all vertices are well connected to the other side. Otherwise, separate the vertices with higher degrees, then find copies of $K_{1,s}$, and afterwards find additional copies of $K_{1,r}$, such that the leaves are well connected.\n    \\item \\label{stepe:absorb} Absorb these copies of $K_{1,s}$ and $K_{1,r}$ into an $r$-regular path-structure and then connect these together into one longer path-structure.\n    After removing the path that we constructed we are left with sets $A_1\\subseteq A$ and $B_1 \\subseteq B$ with $|A_1|=|B_1|$.\n    \\item \\label{stepe:absorb2} Absorb all vertices that do not have large degree to the other side into the path by alternating between both sides.\n    After removing these vertices we are left with sets $A_2\\subseteq A_1$ and $B_2 \\subseteq B_1$ with $|A_2|=|B_2|$ and the property that all vertices have large degree to the other side.\n    \\item \\label{stepe:cover} It is easy to see that $(A_2,B_2)$ is a super-regular pair and that we can cover it with the $\\tfrac r2$-blow-up of a path using Lemma~\\ref{lem:blowup}.\n\\end{enumerate}\nIf we take care of the end-tuples between each of the steps this gives an $r$-regular $r$-connected path-structure covering $G$.\nFor the first step we use the following.\n\\begin{lemma}\n\\label{lem:stars}\n    For any integer $s$ there exists $\\alpha>0$ such that the following holds.\n    Let $G$ be an $n$ vertex graph with maximum degree $\\Delta(G) \\le 4 s \\alpha n$ and minimum degree $\\delta(G) \\ge m + s - 1$, where $1 \\le m \\le \\alpha n$.\n    Then there are $2m$ pairwise disjoint copies of $K_{1,s}$ in $G$.\n\\end{lemma}\n\nThe proof of this lemma and the second extremal case will be given in Section~\\ref{sec:extremal2}.\n\n\\subsection{Constructions}\n\\label{sec:o_constructions}\nFirst recall that the $\\tfrac r2$-blow-up of a cycle is $r$-regular and also $r$-connected.\nIt will not always be possible to construct this, but it will be the basic building block.\nWe might need to absorb some exceptional vertices, for example, when $n$ is not divisible by $r$.\nIn the case when $r$ is even we then remove a perfect matching from one $K_{s,s}$ and add one vertex that is connected to all $2s=r$ vertices that just lost one neighbour (c.f.~Figures~\\ref{fig:absorber_even},~\\ref{fig:absorbK1s}, and~\\ref{fig:Step54}).\nThe resulting graph is still $r$-connected, because we can not disconnect this part of the cycle by removing less than $\\tfrac r2$ vertices.\nA similar construction will be used in the case when $r$ is odd (c.f.~Figures~\\ref{fig:absorber_odd},~\\ref{fig:absorbK1s}~and~\\ref{fig:Step55}) that also preserves $r$-connectivity.\nApart from this, we also have to connect to $\\tfrac r2$-blow-ups of cycles by using at most $r$ edges between them (c.f.~Step~\\ref{stepd:bridge} of Section~\\ref{sec:o_extremal1}).\nWe will only need to take care of a small linear fraction of the vertices from $G$ and, therefore, almost all vertices are in the $\\tfrac r2$-blow-up of a path.\n\n\\@ifstar{\\origsection*}{\\mysection}{Extremal Case I}\n\\label{sec:extremal1}\nIn this section we deal with the first extremal case.\nWe will not use the regularity lemma in this part, but the blow-up lemma will still be helpful.\n\n\\begin{proof}[Proof of Extremal Case I]\nLet $r \\geq 3$ be an integer, let $\\eps>0$ be given by Lemma~\\ref{lem:blowup} on input $\\tfrac 12$, $\\tfrac 12$, and $r$ and let $0<\\alpha \\le \\eps (1000 r^2)^{-1}$.\nLet $G$ be an $n$-vertex graph with $\\delta(G) \\geq \\tfrac {n+r-2}{2}$ and let $A,B \\subseteq V(G)$ with $(\\tfrac 12-\\alpha)n \\le |A|,|B| \\le (\\tfrac 12+\\alpha)n$ such that $G[A]$ and $G[B]$ have minimum degree $(\\tfrac 12-3\\alpha)n$ and every vertex in $C=V(G) \\setminus (A \\cup B)$ has degree at least $\\alpha n$ into $A$ and $B$.\nOur goal is to find an $r$-regular, $r$-connected spanning subgraph in $G$ provided that $n$ is large enough.  \n\n\\subsection{The even case}\n\nAssume that $r$ is even.\nWe begin by constructing $\\tfrac r2$ bridges of size 2 between $A$ and $B$ (\\ref{stepd:bridge} of Section \\ref{sec:o_extremal1}). A visualisation can be found in Figure~\\ref{fig:bridge_graph_h}.\n\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}\n        \\node[circle, draw=black] (a_11) at (0, 3) {$a_{11}$};\n        \\node[circle, draw=black] (a_12) at (0, 1.5) {$a_{12}$};\n        \\node[circle, draw=black] (b_11) at (0, -1.5) {$b_{11}$};\n        \\node[circle, draw=black] (b_12) at (0, -3) {$b_{12}$};\n        \n        \\node[circle, draw=black] (a_21) at (2, 3) {$a_{21}$};\n        \\node[circle, draw=green] (xa1) at (2, 1.5) {$x_{a_1}$};\n        \\node[circle, draw=green] (xb1) at (2, -1.5) {$x_{b_1}$};\n        \\node[circle, draw=black] (b_22) at (2, -3) {$b_{22}$};\n        \n        \\node[circle, draw=black] (a_31) at (4, 3) {$a_{31}$};\n        \\node[circle, draw=green] (xa2) at (4, 1.5) {$x_{a_2}$};\n        \\node[circle, draw=green] (xb2) at (4, -1.5) {$x_{b_2}$};\n        \\node[circle, draw=black] (b_32) at (4, -3) {$b_{32}$};\n        \n        \\node[circle, draw=black] (a_41) at (6, 3) {$a_{41}$};\n        \\node[circle, draw=black] (a_42) at (6, 1.5) {$a_{42}$};\n        \\node[circle, draw=black] (b_41) at (6, -1.5) {$b_{41}$};\n        \\node[circle, draw=black] (b_42) at (6, -3) {$b_{42}$};\n        \n   \n        \\path[-] (a_11) edge[draw=black] (a_21);\n        \\path[-] (a_12) edge[draw=black] (xa1);\n        \\path[-] (a_11) edge[draw=black] (xa1);\n        \\path[-] (a_12) edge[draw=black] (a_21);\n        \n        \\path[-] (xa1) edge[draw=black] (a_31);\n        \\path[-] (a_21) edge[draw=black] (xa2);\n        \\path[-] (a_21) edge[draw=black] (a_31);\n        \n        \\path[-] (a_31) edge[draw=black] (a_41);\n        \\path[-] (xa2) edge[draw=black] (a_42);\n        \\path[-] (a_31) edge[draw=black] (a_42);\n        \\path[-] (xa2) edge[draw=black] (a_41);\n        \n \n        \\path[-] (b_11) edge[draw=black] (xb1);\n        \\path[-] (b_12) edge[draw=black] (b_22);\n        \\path[-] (b_11) edge[draw=black] (b_22);\n        \\path[-] (b_12) edge[draw=black] (xb1);\n        \n        \\path[-] (b_22) edge[draw=black] (xb2);\n        \\path[-] (xb1) edge[draw=black] (b_32);\n        \\path[-] (b_22) edge[draw=black] (b_32);\n        \n        \\path[-] (xb2) edge[draw=black] (b_41);\n        \\path[-] (b_32) edge[draw=black] (b_42);\n        \\path[-] (xb2) edge[draw=black] (b_42);\n        \\path[-] (b_32) edge[draw=black] (b_41);\n        \n        \\path[-] (xb2) edge[draw=green] (xa2);\n        \\path[-] (xb1) edge[draw=green] (xa1);\n\n    \\end{tikzpicture}\n    \\caption{Bridge between the sets $A$ and $B$ in the special case $r = 4$.}\n    \\label{fig:bridge_graph_h}\n\\end{figure}\n\n\n\\begin{claim}\\label{lemma_matching_edges}\nSuppose $\\delta(G) \\geq \\tfrac {n+r-2}{2}$ and $\\abs{A} \\leq \\abs{B}$. There is a matching $(x_{a_1}x_{b_1}, \\ldots, x_{a_r}x_{b_r})$  such that $\\abs{N(x_{a_i}) \\cap A} \\geq \\tfrac n5$ and $\\abs{N(x_{b_j}) \\cap B} \\geq \\tfrac n5$ for all $i, j \\leq r$.\n\\end{claim}\n\\begin{claimproof}\n In order to construct the matching, it suffices to find $r$ edges from $A$ to $V \\setminus A$. Indeed, suppose we find the $r$ edges $a_1c_1, \\ldots, a_rc_r$. If $c_i \\in B$, we take this edge. If $c_i \\in V \\setminus (A \\cup B)$, $c_i$ has either $\\tfrac n5$ edges into $B$ (in this case, take edge $a_ic_i$), or it has $\\tfrac n5$ edges into $A$. Let $i_1, \\ldots, i_l$ be the indices such that $c_{i_j}$ does not have $\\tfrac n5$ edges into $B$. By definition of $\\alpha$-extremity, each $c_{i_j}$ has $\\alpha n$ neighbors in $B$. Select $b_{i_1}, \\ldots, b_{i_l}$ s.t. $b_{i_j} \\in N(c_{i_j}) \\cap B$ and $b_{i_j} \\neq b_{i_k}$ for all $j \\neq k$ (and being disjoint from those $c_i \\in B$ (this is clearly possible). We add edges $b_{i_j}c_{i_j}$ to the matching. \n \nIt remains to show that these edges exist. \nFirst, suppose that $n$ is even. If $\\abs{A} \\leq \\tfrac{n-r}{2}$, the minimum degree of $ \\tfrac{n+r-2}{2}$ guarantees that each vertex of $A$ needs to find at least $\\tfrac{n+r-2}{2} - (\\tfrac{n-r}{2} - 1) = r$ neighbors outside of $A$, hence the assertion follows.\nSuppose $\\abs{A} = \\tfrac{n-r}{2} + i$ with $i = 1, \\ldots, \\tfrac r2$. In this case, $\\abs{V \\setminus A} = \\tfrac{n+r}{2} - i$. Each vertex of $A$ finds at least $r-i$ neighbors outside of $A$. Suppose that $N(A) \\setminus A$ has size at most $r-1$ (thus, all edges from $A$ into the rest of the graph belong to $r-1$ vertices). Now pick a different vertex in the complement (which exists, as $\\tfrac{n+r}{2} - i \\gg r$). This vertex requires $\\tfrac{n+r-2}{2} - (\\tfrac{n+r}{2} - i - 1) = i$ neighbors in $A$, which is a contradiction.\n\nNow, if $n$ is odd, because the minimum degree needs to be an integer, it is at least $\\tfrac{n+1+r-2}{2}$, hence upon removal of one vertex, we are left with a graph on $n' = n-1$ vertices and minimum degree at least $\\tfrac{n+1+r-2}{2}-1 = \\tfrac{n'+r-2}{2}$ (and $n'$ being even). Hence the assertion follows from the previous discussion.\n\\end{claimproof}\n\nTherefore, Claim~\\ref{lemma_matching_edges} gives us the green sub-structure of Figure \\ref{fig:bridge_graph_h}. \nNow, we take two of those matching edges (think of them as being $\\tfrac r2$ pairs of $2$ edges). Denote the vertices that are connected to at least $\\tfrac n5$ vertices in $A$ as $x_{a_1}, x_{a_2}$. We next prove that the black structure around $x_{a_1}, x_{a_2}$ shown Figure \\ref{fig:bridge_graph_h} exists.\n\n\\begin{claim}\\label{bridge_graph_existence}\nThere are distinct vertices $a_{i,1}, \\ldots, a_{i, r\/2} \\in A$ for $i=1,4$ and $a_{i,1}, \\ldots, a_{i, r\/2-1} \\in A$ for $i=2,3$ with the following properties.\n\\begin{enumerate}\n    \\item The edges $a_{i, j}a_{i+1, k}$ for $i=1, 2 ,3$ and $j = 1 , \\ldots, \\tfrac r2$ (or $\\tfrac r2 - 1$, respectively) exist,\n    \\item the edges $x_{a_1}a_{1, j}$ and $x_{a_2}a_{4, j}$ exist for $j = 1 , \\ldots,\\tfrac r2$,\n    \\item the edges $x_{a_1}a_{3, j}$ and $x_{a_2}a_{1, j}$ exist for $j = 1 , \\ldots,\\tfrac r2 -1$.\n\\end{enumerate}\n\\end{claim}\n\n\\begin{claimproof}\nWe select $r - 1$ vertices $a_{1,i}$, $a_{3, j} \\in N(x_{a_1}) \\cap A$ arbitrarily (but disjoint from $x_{a_2}$). Those exist as $x_{a_1}$ has at least $\\tfrac n5$ neighbors in $A$. Each of those vertices is connected to at least $(\\tfrac 12 - 3 \\alpha)n$ vertices in $A$, hence each vertex has at least $(\\tfrac 12 - 3 \\alpha)n - r - 1$ neighbors in $A$ that do not belong to $a_{1,i}$, $a_{3, j}$ or $x_{a_1}, x_{a_2}$.\nTherefore, the joint neighborhood \n\\[N := \\bc{A \\cap \\bigcap_{i = 1}^{r\/2} N(a_{1,i}) \\bigcap_{j=1}^{r\/2 - 1} N(a_{1,j})} \\setminus \\bc{\\bigcup_{i=1}^{r\/2} \\cbc{ a_{4, i} } \\bigcup_{j=1}^{r\/2-1} a_{3, j} \\cup \\cbc{ x_{a_1}, x_{a_2} }}\\]\nhas size at least $(\\tfrac 12 - 4 r \\alpha)n$. Therefore, we find \n\\[\\abs{ N(x_{a_2}) \\cap N } \\geq \\frac{n}{100} \\, , \\]\nthus the claim follows as the same token holds in $B$ as well.\n\\end{claimproof}\nWe denote the resulting collection of vertices in $A$ by $X_{A, 1}, \\ldots, X_{A, r\/2}$.\nClearly, each vertex in $A$ stays connected to at least $(\\tfrac 12 - 4 \\alpha)n$ vertices in\n\\[A'_{0} =  A \\setminus \\bc{ V(X_{A, 1}) \\cup \\ldots \\cup V(X_{A, r\/2}) } \\, .\\]\n\nNext, we introduce a \\textit{gluing operation} GE.\n\\begin{claim}[Gluing operation GE]\nGiven two disjoint sets $D_1, D_2 \\subset A'_0$ of size exactly $\\tfrac r2$, we find two disjoint sets $D, D' \\subset A'_0 \\setminus (D_1 \\cup D_2)$ of size $\\tfrac r2$ such that\n\\[G[D_1, D] \\equiv K_{r\/2, r\/2}, \\quad G[D, D'] \\equiv K_{r\/2, r\/2}\\quad \\text{and} \\quad G[D', D_2] \\equiv K_{r\/2, r\/2} \\, .\\]\n\\end{claim}\n\\begin{claimproof}\nAs the joint $A'_0$ - neighborhood of $D_1$ and $D_2$ has size at least $(\\tfrac 12 - 10r \\alpha)n$, the assertion follows.\n\\end{claimproof}\n\nUsing GE, we glue the $\\tfrac r2$ bridges in $A$ and $B$ respectively together using mutually disjoint vertex sets $D^A_1, \\ldots,D^A_{r\/2 - 1}$ and $D^B_1, \\ldots,D^B_{r\/2 - 1}$ and are left with path-like structures $P_A$ and $P_B$. After gluing the bridges together, we let $A' = A'_0 \\setminus V(P_A), B' = B'_0 \\setminus V(P_B)$. \n\nIn a next step we need to absorb left-over vertices (\\ref{stepd:absorb} of Section \\ref{sec:o_extremal1}). To this end define two absorber-graphs for a vertex $u$: $\\xi_r(u)$ and $\\xi'_r(u)$ (see Figure \\ref{fig:absorber_even}).\n\\begin{definition}\n    Let $D \\in \\cbc{A', B'}$ and $u$ a vertex such that $\\abs{ N(u) \\cap D } \\geq \\tfrac n6$. Define $\\xi_r(u)$ as follows.\n    \\begin{itemize}\n        \\item Select $D_1 = \\cbc{d_1, d_2, \\ldots, d_{r\/2}}, D_2 = \\cbc{d'_1, \\ldots, d'_{r\/2}} \\subset N(u) \\cap D$, hence $r$ pairwise disjoint vertices.\n        \\item Select $D' = \\cbc{ u'_1, \\ldots, u'_{r\/2 - 1}} \\subset N(D_1) \\cap N(D_2) \\cap D \\setminus (D_1 \\cup D_2 \\cup \\cbc{u}) $.\n    \\end{itemize}\n    Define $\\xi_r(u)$ as the graph containing $D_1, D_2, D'$ and $u$ as well as all the edges from $D_1$ to $D' \\cup \\cbc{u}$ and from $D_2$ to $D' \\cup \\cbc{u}$.\n    Furthermore, define $\\xi'_r(u)$ via\n    \\begin{itemize}\n        \\item Select $D_1 = \\cbc{d_1, d_2, \\ldots, d_{r\/2}}, D_2 = \\cbc{d'_1, \\ldots, d'_{r\/2}} \\subset N(u) \\cap D$, hence $r$ pairwise disjoint vertices.\n        \\item Select $D' = \\cbc{ u'_1, \\ldots, u'_{r\/2 - 1}} \\subset N(D_1) \\cap N(D_2) \\cap D \\setminus (D_1 \\cup D_2 \\cup \\cbc{u}) $.\n        \\item Select $\\tfrac r2$ vertices $E_0 = \\cbc{e_0, \\ldots, e_{r\/2-1}}$ and an additional disjoint vertex $e_{r\/2}$ from $D \\setminus \\bc{ D_1 \\cup D_2 \\cup D' }$ such that $E_0 \\cup \\cbc{e_2} \\subset N(d_2) \\cap \\ldots \\cap N(d_{r\/2}) \\cap D$ and $G[E_0, \\cbc{e_{r\/2}}] \\equiv K_{r\/2, 1}$. \n    \\end{itemize}\n    Define $\\xi'_r(u)$ as the graph containing $D_1, D_2, D'$ and $u$ as well as all the edges from $D_1$ to $D' \\cup \\cbc{u}$ and from $D_2$ to $D' \\cup \\cbc{u}$.\n\\end{definition}\nClearly, by the sizes of $A', B'$ and the minimum-degree condition, given at most $100 \\alpha n$ pairwise different vertices, there is an absorber for each vertex which is disjoint from all other absorbers. Furthermore, by the minimum degree condition inside of $A'$ and $B'$, this family of absorbers exists.\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}\n    \n\n     \\node[circle, draw=green!40] (d1) at (2, 1) {$d_1$};\n     \\node[circle, draw=green!40] (d2) at (2, -1) {$d_2$};\n\n     \\node[circle, draw=green!40] (dp1) at (6, 1) {$d'_{1}$};\n     \\node[circle, draw=green!40] (dp2) at (6, -1) {$d'_{2}$};\n     \n     \\node[circle, draw=blue!40] (up1) at (4, 1) {$u'_{1}$};\n     \\node[circle, draw=green] (u) at (4, -1) {$u$};\n\n     \n     \\path[-] (u) edge[draw=green] (d1);\n     \\path[-] (u) edge[draw=green] (d2);\n     \\path[-] (u) edge[draw=green] (dp1);\n     \\path[-] (u) edge[draw=green] (dp2);\n     \n     \\path[-] (d1) edge[draw=blue] (up1);\n     \\path[-] (d2) edge[draw=blue] (up1);\n     \\path[-] (dp1) edge[draw=blue] (up1);\n     \\path[-] (dp2) edge[draw=blue] (up1);\n     \n    \n     \n     \\node[circle, draw=blue!40] (e0) at (10, 1) {$e_0$};\n     \\node[circle, draw=blue!40] (e1) at (10, -1) {$e_1$};\n     \\node[circle, draw=blue!40] (e2) at (11, 0) {$e_2$};\n     \n     \\node[circle, draw=green!40] (ld1) at (12, 1) {$d_1$};\n     \\node[circle, draw=green!40] (ld2) at (12, -1) {$d_2$};\n\n     \\node[circle, draw=green!40] (ldp1) at (16, 1) {$d'_{1}$};\n     \\node[circle, draw=green!40] (ldp2) at (16, -1) {$d'_{2}$};\n     \n     \\node[circle, draw=blue!40] (lup1) at (14, 1) {$u'_{1}$};\n     \\node[circle, draw=green] (lu) at (14, -1) {$u$};\n\n     \n     \\path[-] (lu) edge[draw=green] (ld1);\n     \\path[-] (lu) edge[draw=green] (ld2);\n     \\path[-] (lu) edge[draw=green] (ldp1);\n     \\path[-] (lu) edge[draw=green] (ldp2);\n     \n     \\path[-] (ld1) edge[draw=blue] (lup1);\n     \\path[-] (ld2) edge[draw=blue] (lup1);\n     \\path[-] (ldp1) edge[draw=blue] (lup1);\n     \\path[-] (ldp2) edge[draw=blue] (lup1);\n     \n     \\path[-] (ld1) edge[draw=blue] (e2);\n     \\path[-] (ld2) edge[draw=blue] (e2);\n     \\path[-] (e1) edge[draw=blue] (e2);\n     \\path[-] (e0) edge[draw=blue] (e2);\n     \n     \\path[-] (e0) edge[draw=blue, bend left=30] (ld2);\n     \\path[-] (e1) edge[draw=blue, bend left=30] (ld1);\n    \n    \\end{tikzpicture}\n    \n    \\caption{Absorbers $\\xi_4(u)$ (left) and $\\xi'_4(u)$ (right) with $r=4$, where $u$ is the green vertex, the green vertices are inside the $A'$-neighborhood of $u$ and the blue vertices are vertices chosen from $A'$.}\n    \\label{fig:absorber_even}\n\\end{figure}\n\nWe are now in position to absorb the exceptional set $C$ (of course, without the bridging vertices on $P_A$ and $P_B$. For all up to $6 \\alpha n$ vertices in $C$ create a disjoint absorber $\\xi_r(u)$ as above. Chose as $D$ either $A'$ (if a vertex has $\\tfrac n5$ neighbors in $A$), or $B'$ otherwise. Next, using the gluing operation GE up to $6 \\alpha n$ times, glue the absorbers inside of $A'$ and $B'$ together, always using only vertices that did not get used in a previous gluing step or are part of the absorbers. As one only requires at most $24 r \\alpha n$ vertices during this procedure, it is clearly possible by above discussion.\n\nFinally, use GE again to glue the series of absorbers to $P_A$ and $P_B$ respectively (which is clearly possible, as this is only one operation on each set).\n\nAfter the repetitive gluing, we are left with sets $A''$ and $B''$ (hence, $A'$ without the glued structures $P'_A$) and the path-like subgraph $P'_A$ of size at most $25r \\alpha n$, hence $\\abs{A''} \\geq (\\tfrac 12 - 30 r \\alpha)n$. Furthermore, each vertex in $A''$ is connected to at least $(\\tfrac 12 - 40 r \\alpha)n$ vertices in $A''$. Clearly, the same holds for $B''$. \n\nBefore closing the path in both sets (hence, creating a cycle which contains all vertices that are not part of $P'_A$ or $P'_B$), which is a standard application of the blow-up lemma, we need to make sure that certain divisibility conditions hold. \nAs we wish to close the cycle by appending blocks of two layers of size $r$ (thus, $K_{{r\/2}, {r\/2}}$), we require that $\\abs{A''} \\equiv \\abs{B''} \\equiv 0 \\pmod r$. If this is the case, set $A''' = A''$ and proceed. Otherwise, if there is $0 < i < r$ such that $\\abs{A''} \\equiv i \\pmod r$, select $i$ vertices $a_1, \\ldots,a_{i} \\in A''$ and absorb them using disjoint instances $\\xi'_r(a_1), \\ldots, \\xi'_r(a_{i})$ with $D = A''$. Clearly, as this requires only finitely many vertices, such a disjoint family exists. Further, because $a_j \\in A''$, each absorber consumes $2r + 1$ vertices of $A''$, hence afterwards, the divisibility condition holds. Now, glue the absorbers sequentially to $P'_A$ using GE and sets $G_1, \\ldots,G_{i-1}$. As each gluing operation consumes $r$ vertices, the divisibility does not change hence we are left with a set $A''' = A'' \\setminus \\bc{ V(\\xi'_r(a_1)) \\cup \\ldots \\cup V(\\xi'_r(a_{i})) \\cup G_1 \\cup \\ldots \\cup G_{i-1} }$.\n\nNow it is easy to check that $(A''',B''')$ is $(\\eps,\\tfrac 12)$-super-regular and by Lemma~\\ref{lem:blowup} we find an $\\tfrac r2-$blowup of the path on all remaining vertices of $A'''$ and $B'''$ (\\ref{stepd:cover} of Section \\ref{sec:o_extremal1}). \nMoreover, the end-tuples of the path-like structure constructed before have at least $\\tfrac 12 |A'''|$ and $\\tfrac 12 |B'''|$ common neighbours in $A'''$ and $B'''$ respectively.\nTherefore, we may choose the start- and end-tuples of this path-blow-up to connect to these end-tuples.\n\nWe are left to argue that the constructed subgraph is $r$-connected and $r$-regular.\n\\begin{claim}\nThe constructed subgraph is $r$-connected and $r$-regular.\n\\end{claim}\n\\begin{claimproof}\nWhile $r$-regularity follows obviously, the $r$-connected part needs a short argument. Upon removal of up to $r-1$ bridge-vertices, the parts do not fall apart. Furthermore, removing up to $r-1$ vertices in the $\\tfrac r2$-blow-up of the path part of the subgraph does not disconnect the structure. Finally, the absorbing structure $\\xi_r$ itself is isomorphic to an $\\tfrac r2$-blowup of the path on three vertices. Moreover, disconnecting the graph by removing up to $r-1$ vertices in $\\xi_r'$ is not possible.\n\\end{claimproof}\n\n\n\n\\subsection{The odd case}\n\n\nAssume that $r$ is odd.\nThe argument in the odd case is a bit more delicate as in the even case. Indeed, while in the process above all divisibility conditions could be easily established, in the odd case, we might end with two almost cliques of odd size. If there is a set $C$, we can easily absorb those vertices in a way that after absorbing both parts of the graph contain an even number of vertices - which we require to embed a regular graph. If on the other hand there is no such set $C$, we need to be much more careful. We will tackle this problem by having two different types of bridges between $A$ and $B$, one consuming an even number of vertices of each set, one consuming an odd number - thus, depending on the size of $C$ and the parity of $A$ and $B$, we need to use two different constructions. The two types of bridges are visualised in Figure~\\ref{fig:bridge_graph_h_odd} for the special case $ r = 5$.\n\n\n\\begin{figure}\n    \\centering\n    \\begin{minipage}{0.45\\textwidth}\n    \\begin{tikzpicture}\n        \\node[circle, draw=black] (a_11) at (0, 4.5) {$a_{11}$};\n        \\node[circle, draw=black] (a_12) at (0, 3) {$a_{12}$};\n        \\node[circle, draw=black] (a_13) at (0, 1.5) {$a_{13}$};\n        \\node[circle, draw=black] (b_11) at (0, -1.5) {$b_{11}$};\n        \\node[circle, draw=black] (b_12) at (0, -3) {$b_{12}$};\n        \\node[circle, draw=black] (b_13) at (0, -4.5) {$b_{13}$};\n        \n        \\node[circle, draw=black] (a_21) at (2, 4.5) {$a_{21}$};\n        \\node[circle, draw=black] (a_22) at (2, 3) {$a_{22}$};\n        \\node[circle, draw=green] (xa1) at (2, 1.5) {$x_{a_1}$};\n        \\node[circle, draw=green] (xb1) at (2, -1.5) {$x_{b_1}$};\n        \\node[circle, draw=black] (b_22) at (2, -3) {$b_{22}$};\n        \\node[circle, draw=black] (b_23) at (2, -4.5) {$b_{23}$};\n        \n        \\node[circle, draw=black] (a_31) at (4, 4.5) {$a_{31}$};\n        \\node[circle, draw=black] (a_32) at (4, 3) {$a_{32}$};\n        \\node[circle, draw=green] (xa2) at (4, 1.5) {$x_{a_2}$};\n        \\node[circle, draw=green] (xb2) at (4, -1.5) {$x_{b_2}$};\n        \\node[circle, draw=black] (b_32) at (4, -3) {$b_{32}$};\n        \\node[circle, draw=black] (b_33) at (4, -4.5) {$b_{33}$};\n        \n        \\node[circle, draw=black] (a_41) at (6, 4.5) {$a_{41}$};\n        \\node[circle, draw=black] (a_42) at (6, 3) {$a_{42}$};\n        \\node[circle, draw=black] (a_43) at (6, 1.5) {$a_{43}$};\n        \\node[circle, draw=black] (b_41) at (6, -1.5) {$b_{41}$};\n        \\node[circle, draw=black] (b_42) at (6, -3) {$b_{42}$};\n        \\node[circle, draw=black] (b_43) at (6, -4.5) {$b_{43}$};\n        \n   \n       \n        \\path[-] (a_11) edge[draw=black] (a_22);\n        \\path[-] (a_11) edge[draw=black] (xa1);\n        \\path[-] (a_12) edge[draw=black] (xa1);\n        \\path[-] (a_12) edge[draw=black] (a_21);\n       \n        \\path[-] (a_13) edge[draw=black] (a_21);\n        \\path[-] (a_13) edge[draw=black] (a_22);\n\n        \\path[-] (xa1) edge[draw=black] (a_31);\n        \\path[-] (xa1) edge[draw=black] (a_32);\n        \\path[-] (a_21) edge[draw=black] (xa2);\n        \\path[-] (a_21) edge[draw=black] (a_31);\n        \\path[-] (a_21) edge[draw=black] (a_32);\n        \\path[-] (a_22) edge[draw=black] (xa2);\n        \\path[-] (a_22) edge[draw=black] (a_31);\n        \\path[-] (a_22) edge[draw=black] (a_32);\n        \n       \n        \\path[-] (a_31) edge[draw=black] (a_42);\n        \\path[-] (a_31) edge[draw=black] (a_43);\n        \\path[-] (a_32) edge[draw=black] (a_41);\n       \n        \\path[-] (a_32) edge[draw=black] (a_43);\n        \\path[-] (xa2) edge[draw=black] (a_41);\n        \\path[-] (xa2) edge[draw=black] (a_42);\n       \n        \n \n       \n       \n        \\path[-] (b_12) edge[draw=black] (b_23);\n        \\path[-] (b_11) edge[draw=black] (b_22);\n        \\path[-] (b_11) edge[draw=black] (b_23);\n        \\path[-] (b_12) edge[draw=black] (xb1);\n        \\path[-] (b_13) edge[draw=black] (b_22);\n       \n        \\path[-] (b_13) edge[draw=black] (xb1);\n        \n        \n        \\path[-] (b_22) edge[draw=black] (xb2);\n        \\path[-] (xb1) edge[draw=black] (b_32);\n        \\path[-] (xb1) edge[draw=black] (b_33);\n        \\path[-] (b_22) edge[draw=black] (b_32);\n        \\path[-] (b_22) edge[draw=black] (b_33);\n        \\path[-] (b_23) edge[draw=black] (b_32);\n        \\path[-] (b_23) edge[draw=black] (xb2);\n        \\path[-] (b_23) edge[draw=black] (b_33);\n        \n       \n        \\path[-] (xb2) edge[draw=black] (b_43);\n       \n        \\path[-] (b_32) edge[draw=black] (b_43);\n        \\path[-] (xb2) edge[draw=black] (b_42);\n        \\path[-] (b_32) edge[draw=black] (b_41);\n        \\path[-] (b_33) edge[draw=black] (b_41);\n        \\path[-] (b_33) edge[draw=black] (b_42);\n       \n        \n        \\path[-] (xb2) edge[draw=green] (xa2);\n        \\path[-] (xb1) edge[draw=green] (xa1);\n\n    \\end{tikzpicture}\n    \\end{minipage} \\hfill\\vline\\hfill\n    \\begin{minipage}{0.45\\textwidth}\n        \\begin{tikzpicture}\n        \\node[circle, draw=black] (a_11) at (0, 4.5) {$a_{11}$};\n        \\node[circle, draw=black] (a_12) at (0, 3) {$a_{12}$};\n        \\node[circle, draw=black] (a_13) at (0, 1.5) {$a_{13}$};\n        \\node[circle, draw=black] (b_11) at (0, -1.5) {$b_{11}$};\n        \\node[circle, draw=black] (b_12) at (0, -3) {$b_{12}$};\n        \\node[circle, draw=black] (b_13) at (0, -4.5) {$b_{13}$};\n        \n        \\node[circle, draw=black] (a_21) at (2, 4.5) {$a_{21}$};\n        \\node[circle, draw=black] (a_22) at (2, 3) {$a_{22}$};\n        \\node[circle, draw=black] (a_23) at (2, 1.5) {$a_{23}$};\n        \\node[circle, draw=black] (b_21) at (2, -1.5) {$b_{21}$};\n        \\node[circle, draw=black] (b_22) at (2, -3) {$b_{22}$};\n        \\node[circle, draw=black] (b_23) at (2, -4.5) {$b_{23}$};\n        \n        \\node[circle, draw=black] (a_31) at (4, 4.5) {$a_{31}$};\n        \\node[circle, draw=black] (a_32) at (4, 3) {$a_{32}$};\n        \\node[circle, draw=black] (a_33) at (4, 1.5) {$a_{33}$};\n        \\node[circle, draw=black] (b_31) at (4, -1.5) {$b_{31}$};\n        \\node[circle, draw=black] (b_32) at (4, -3) {$b_{32}$};\n        \\node[circle, draw=black] (b_33) at (4, -4.5) {$b_{33}$};\n        \n        \\node[circle, draw=black] (a_41) at (6, 4.5) {$a_{41}$};\n        \\node[circle, draw=black] (a_42) at (6, 3) {$a_{42}$};\n        \\node[circle, draw=black] (a_43) at (6, 1.5) {$a_{43}$};\n        \\node[circle, draw=black] (b_41) at (6, -1.5) {$b_{41}$};\n        \\node[circle, draw=black] (b_42) at (6, -3) {$b_{42}$};\n        \\node[circle, draw=black] (b_43) at (6, -4.5) {$b_{43}$};\n        \n        \\node[circle, draw=green] (xa) at (3, 2.25) {$x_{a}$};\n        \\node[circle, draw=green] (xb) at (3, -2.25) {$x_{b}$};\n        \n        \n       \n        \\path[-] (a_12) edge[draw=black] (a_23);\n        \\path[-] (a_11) edge[draw=black] (a_23);\n       \n        \\path[-] (a_11) edge[draw=black] (a_22);\n       \n       \n        \\path[-] (a_12) edge[draw=black] (a_21);\n       \n        \\path[-] (a_13) edge[draw=black] (a_21);\n        \\path[-] (a_13) edge[draw=black] (a_22);\n\n\n        \\path[-] (a_21) edge[draw=black] (a_31);\n        \\path[-] (a_21) edge[draw=black] (a_32);\n        \\path[-] (a_22) edge[draw=black] (a_31);\n        \\path[-] (a_22) edge[draw=black] (a_32);\n        \n       \n        \\path[-] (a_31) edge[draw=black] (a_42);\n        \\path[-] (a_31) edge[draw=black] (a_43);\n        \\path[-] (a_32) edge[draw=black] (a_41);\n       \n        \\path[-] (a_32) edge[draw=black] (a_43);\n        \\path[-] (xa2) edge[draw=black] (a_41);\n        \\path[-] (xa2) edge[draw=black] (a_42);\n       \n        \n \n       \n       \n        \\path[-] (b_12) edge[draw=black] (b_23);\n        \\path[-] (b_11) edge[draw=black] (b_22);\n        \\path[-] (b_11) edge[draw=black] (b_23);\n        \\path[-] (b_12) edge[draw=black] (xb1);\n        \\path[-] (b_13) edge[draw=black] (b_22);\n       \n        \\path[-] (b_13) edge[draw=black] (xb1);\n        \n        \n       \n      \n       \n        \\path[-] (b_22) edge[draw=black] (b_32);\n        \\path[-] (b_22) edge[draw=black] (b_33);\n        \\path[-] (b_23) edge[draw=black] (b_32);\n     \n        \\path[-] (b_23) edge[draw=black] (b_33);\n        \n       \n      \n       \n        \\path[-] (b_32) edge[draw=black] (b_43);\n      \n        \\path[-] (b_32) edge[draw=black] (b_41);\n        \\path[-] (b_33) edge[draw=black] (b_41);\n        \\path[-] (b_33) edge[draw=black] (b_42);\n       \n        \n        \\path[-] (b_31) edge[draw=black] (b_42);\n        \\path[-] (b_31) edge[draw=black] (b_43);\n        \n        \\path[-] (b_21) edge[draw=black, bend right=10] (b_33);\n        \\path[-] (b_31) edge[draw=black, bend left=10] (b_23);\n        \\path[-] (b_31) edge[draw=black] (b_21);\n        \n        \\path[-] (xb) edge[draw=black] (b_21);\n        \\path[-] (xb) edge[draw=black] (b_22);\n        \\path[-] (xb) edge[draw=black] (b_31);\n        \\path[-] (xb) edge[draw=black] (b_32);\n        \n        \\path[-] (a_23) edge[draw=black, bend left=10] (a_31);\n        \\path[-] (a_33) edge[draw=black, bend right=10] (a_21);\n        \\path[-] (a_23) edge[draw=black] (a_33);\n        \n        \\path[-] (xa) edge[draw=black] (a_22);\n        \\path[-] (xa) edge[draw=black] (a_32);\n        \\path[-] (xa) edge[draw=black] (a_23);\n        \\path[-] (xa) edge[draw=black] (a_33);\n        \n        \n        \\path[-] (xa) edge[draw=green] (xb);\n     \n\n    \\end{tikzpicture}\n    \\end{minipage}\n    \\caption{The two types of connections between the sets $A$ and $B$ in the special case $r = 5$.}\n    \\label{fig:bridge_graph_h_odd}\n\\end{figure}\n\nWe begin by showing that we find three pairs of bridges of the first type, using an even number of vertices of both classes (\\ref{stepd:bridge} of Section \\ref{sec:o_extremal1}).\n\n\\begin{claim}\\label{lemma_matching_edges_odd}\nSuppose $\\delta(G) \\geq \\tfrac{n+r-2}{2}$ and $\\abs{A} \\leq \\abs{B}$. Furthermore, let $n$ be large enough. There is a matching $(x_{a_1}x_{b_1}, \\ldots, x_{a_{r+1}}x_{b_{r+1}})$  such that $\\abs{N(x_{a_i}) \\cap A} \\geq \\tfrac n5$ and $\\abs{N(x_{b_j}) \\cap B} \\geq \\tfrac n5$ for all $i, j \\leq r+1$.\n\\end{claim}\n\\begin{claimproof}\nAs in the proof of Claim~\\ref{lemma_matching_edges}, it suffices to find $r+1$ edges from $A$ to $V \\setminus A$. \n\nFirst, suppose that $n$ is odd. If $\\abs{A} \\leq \\tfrac{n-r-2}{2}$, the minimum degree of $ \\tfrac{n+r-2}{2}$ guarantees that each vertex of $A$ needs to find at least $\\tfrac{n+r-2}{2} - (\\tfrac{n-r-2}{2} - 1) = r+1$ neighbors outside of $A$, hence the assertion follows.\nSuppose $\\abs{A} = \\tfrac{n-r}{2} + i$ with $i = 1, \\ldots,\\floor{\\tfrac r2}$. In this case, $\\abs{V \\setminus A} = \\tfrac{n+r}{2} - i$. Each vertex of $A$ finds at least $r-i$ neighbors outside of $A$. Suppose that $N(A) \\setminus A$ has size at most $r$ (thus, all edges from $A$ into the rest of the graph belong to at most $r$ vertices). Now pick a different vertex in the complement (which exists, as $\\tfrac{n+r}{2} - i \\gg r$). This vertex requires $\\tfrac{n+r-2}{2} - (\\tfrac{n+r}{2} - i - 1) = i$ neighbors in $A$, which is a contradiction.\nIf finally $\\abs{A} = \\tfrac{n-r}{2}$, each vertex of $A$ has at least $r$ neighbors in the complement of $A$. If all vertices of $A$ share those $r$ vertices (hence, we only find $r$ matching edges), those vertices are connected to all vertices in $A$, hence can be moved to $A$ by only increasing $\\alpha$-extremity, thus the assertion follows from the previous case.\n\nNow, if $n$ is even, because the minimum degree needs to be an integer, it is at least $\\tfrac{n+1+r-2}{2}$, hence upon removal of one vertex, we are left with a graph on $n' = n-1$ vertices and minimum degree at least $\\tfrac{n+1+r-2}{2} -1= \\tfrac{n'+r-2}{2}$ (and $n'$ being odd). Hence the assertion follows from the previous discussion.\n\\end{claimproof}\n\nSimilarly as in the even case, Claim~\\ref{lemma_matching_edges_odd} gives us $\\tfrac{r+1}{2}$ pairs of bridge-edges as in Figure \\ref{fig:bridge_graph_h_odd} (the green part). Clearly, the rest of the bridge graph can be created completely analogously to Claim~\\ref{bridge_graph_existence}.\n\nBy the same token we get the following claim immediately.\n\\begin{claim}\\label{cor_matching_edges_odd}\nSuppose $\\delta(G) \\geq \\tfrac{n+r-2}{2}$ and $\\abs{A} \\leq \\abs{B}$. Furthermore, let $n$ be large enough. There is a matching $(x_{a_1}x_{b_1}, \\ldots, x_{a_{r-1}}x_{b_{r-1}})$  such that $\\abs{N(x_{a_i}) \\cap A} \\geq  \\tfrac n5$ and $\\abs{N(x_{b_j}) \\cap B} \\geq \\tfrac n5$ for all $i, j \\leq r-1$. Furthermore, there are different vertices $x_a \\in A, x_b \\in B$ such that $x_ax_b \\in E(G)$, $\\abs{N(x_a) \\cap A} \\geq r-1$ and $\\abs{ N(x_b) \\cap B} \\geq r-1$.\n\\end{claim}\n\\begin{claimproof}\nThis follows directly from the proof of Claim~\\ref{lemma_matching_edges_odd}.\n\\end{claimproof}\n\nNext, we re-define the gluing operation GE to GO as follows.\n\\begin{claim}[Gluing operation GO]\nGiven two disjoint sets $D_1, D_2 \\subset A'_0$ of sizes exactly $\\tfrac{r+1}{2}$, we find two disjoint sets $D'_1, D'_2 \\subset A'_0 \\setminus (D_1 \\cup D_2)$ of size $\\tfrac{r+1}{2}$ such that\n\\[G[D_1, D'_1] \\equiv G[D'_1, D'_2] \\equiv G[D'_2, D_2] \\equiv K_{(r+1)\/2, (r+1)\/2} \\, .\\]\n\\end{claim}\n\\begin{claimproof}\nThis follows directly from the fact that each vertex in $A'_0$ is connected to at least $(\\tfrac 12 - 4 \\alpha)n$ vertices in $A'_0$ and GO uses only finitely many vertices of the neighborhoods.\n\\end{claimproof}\nWe stress at this point that GO can be applied to blocks whose end-vertices $D_1$ have currently degree $\\tfrac{r-1}{2}$ (then we chose all edges from the connecting graphs $K_{(r+1)\/2, (r+1)\/2}$) or degree $\\tfrac{r+1}{2}$ (then we chose the complete bipartite graph between $D'_1$ and $D'_2$ and at the other connections, we remove one matching of size $\\tfrac{r+1}{2}$.\nFurthermore observe, that gluing consumes $r+1$ vertices from the underlying set.\n\nWe proceed as follows. If $\\abs{C} > 0$ or $\\abs{C} = 0$ and $\\abs{A'_0}, \\abs{B'_0} \\equiv 0 \\pmod 2$, we create $\\tfrac{r+1}{2}$ pairs of bridge vertices by Claim~\\ref{lemma_matching_edges_odd}. Otherwise, we create $\\tfrac{r-1}{2}$ pairs of bridge vertices and one additional bridge by Claim~\\ref{cor_matching_edges_odd}. In both cases, we glue the bridges in $A$ and $B$ respectively together using mutually disjoint vertex sets $D^A_1, \\ldots,D^A_{(r-1)\/2}$ and $D^B_1, \\ldots,D^B_{(r-1)\/2}$ constructing $P_A, P_B$ and, similarly as in the even case, we set $A' = A'_0 \\setminus V(P_A), B' = B'_0 \\setminus V(P_B)$. \nClearly, the parity of $A'$ and $B'$ are both even.\n\nNext, we define absorbing structures for the left-over vertices and for absorbing vertices in order to guarantee divisibility (\\ref{stepd:absorb} of Section \\ref{sec:o_extremal1}). They need to be defined slightly differently as in the even case (Figure \\ref{fig:absorber_odd}).\n\n\\begin{definition}\n    Let $D \\in \\cbc{A', B'}$ and $u$ a vertex such that $\\abs{ N(u) \\cap D } \\geq \\tfrac n6$. Define $\\xi_r(u)$ as follows.\n    \\begin{itemize}\n         \\item Select $D_1 = \\cbc{d_1, d_2, \\ldots, d_{(r+1)\/2}}, D_2 = \\cbc{d'_1, \\ldots, d'_{(r+1)\/2}} \\subset  N(u) \\cap D$, hence $r$ pairwise disjoint vertices.\n         \\item Select $D' = \\cbc{ u'_1, \\ldots, u'_{(r-1)\/2}} \\subset N(D_1) \\cap N(D_2) \\cap D \\setminus (D_1 \\cup D_2 \\cup \\cbc{u}) $.\n         \\item Select $E_0 = \\cbc{ e_1, \\ldots, e_{(r+1)\/2} }$ in the joint $D$-neighborhood of $D_1 \\setminus (D_2 \\cup D' \\cup \\cbc{u})$.  \n    \\end{itemize}\n    Define $\\xi_r(u)$ as the graph containing $E_0, D_1, D_2, D'$ and $u$ as well as all the edges from $D_1$ to $D' \\cup \\cbc{u}$. Furthermore, take all edges from $D_2$ to $D' \\cup \\cbc{u}$ removing one matching of size $\\tfrac{r+1}{2}$ and the edges from $E_0$ to $D_1$ removing a matching as well.\n    Furthermore,  for two adjacent vertices $u_1, u_2 \\in D$ define $\\xi'_r(u_1, u_2)$ via\n    \\begin{itemize}\n         \\item Select $F_1 = \\cbc{f_1, f_2, \\ldots, f_{(r+1)\/2}}, F' = \\cbc{f'_1, \\ldots, f'_{(r+1)\/2}} \\subset N(u_1) \\cap N(u_2) \\cap D$, hence $r+1$ pairwise disjoint vertices in the joint neighborhood of $u_1$ and $u_2$.\n         \\item Select the following edges\n         \\begin{itemize}\n             \\item $f_1u_1, \\ldots, f_{(r-1)\/2}u_1$,\n             \\item $f'_1u_1, \\ldots, f'_{(r-1)\/2}u_1$,\n             \\item $f_2u_2, \\ldots, f_{(r+1)\/2}u_2$,\n             \\item $f'_2u_2, \\ldots, f'_{(r+1)\/2}u_2$ and\n             \\item $f_1f'_1$ as well as $f_{(r+1)\/2}f'_{(r+1)\/2}$.\n         \\end{itemize}\n         \\item Finally, draw $(r-2)$ half-edges at each $f_i, f'_i$ and match them such that a simple graph is induced by the matching.\n     \\end{itemize}\n\\end{definition}\nAs $u_1, u_2 \\in D$, hence the neighborhood of $u_1$, $u_2$ contains only vertices that, themselves, have $(\\tfrac 12 - 30 r \\alpha)n$ vertices from $D$, the joint neighborhood of those two vertices has size at least $(\\tfrac 12 - 60 r \\alpha )n$, $\\xi'(u_1, u_2)$ is well defined. Observe that absorbing a vertex $u \\not \\in D$ consumes $2r + 1$ vertices from $D$ while absorbing $u_1, u_2 \\in D$ consumes $r+3$ vertices (including $u_1, u_2$) in $D$.\n\nAs in the even case, we find a family of disjoint structures to absorb at least $100 \\alpha n$ different vertices.\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8]\n    \n    \\node[circle, draw=blue] (e1) at (0, 3) {$e_1$};\n     \\node[circle, draw=blue] (e2) at (0, 1) {$e_2$};\n     \\node[circle, draw=blue] (e3) at (0, -1) {$e_3$};\n\n     \\node[circle, draw=green!40] (d1) at (2, 3) {$d_1$};\n     \\node[circle, draw=green!40] (d2) at (2, 1) {$d_2$};\n     \\node[circle, draw=green!40] (d3) at (2, -1) {$d_3$};\n\n     \\node[circle, draw=green!40] (dp1) at (6, 3) {$d'_{1}$};\n     \\node[circle, draw=green!40] (dp2) at (6, 1) {$d'_{2}$};\n     \\node[circle, draw=green!40] (dp3) at (6, -1) {$d'_{3}$};\n     \n     \\node[circle, draw=blue!40] (up1) at (4, 3) {$u'_{1}$};\n     \\node[circle, draw=blue!40] (up2) at (4, 1) {$u'_{2}$};\n     \\node[circle, draw=green] (u) at (4, -1) {$u$};\n\n     \n     \\path[-] (u) edge[draw=green] (d1);\n     \\path[-] (u) edge[draw=green] (d2);\n     \\path[-] (u) edge[draw=green] (d3);\n     \\path[-] (u) edge[draw=green] (dp1);\n     \\path[-] (u) edge[draw=green] (dp2);\n    \n     \n     \\path[-] (d1) edge[draw=blue] (up1);\n     \\path[-] (d2) edge[draw=blue] (up1);\n     \\path[-] (d1) edge[draw=blue] (up2);\n     \\path[-] (d2) edge[draw=blue] (up2);\n     \\path[-] (d3) edge[draw=blue] (up1);\n     \\path[-] (d3) edge[draw=blue] (up2);\n     \n    \n     \\path[-] (d2) edge[draw=blue] (e1);\n     \\path[-] (d3) edge[draw=blue] (e1);\n     \\path[-] (d1) edge[draw=blue] (e2);\n    \n     \\path[-] (d3) edge[draw=blue] (e2);\n     \\path[-] (d1) edge[draw=blue] (e3);\n     \\path[-] (d2) edge[draw=blue] (e3);\n    \n     \n     \n    \n     \\path[-] (dp2) edge[draw=blue] (up1);\n     \\path[-] (dp3) edge[draw=blue] (up1);\n     \\path[-] (dp1) edge[draw=blue] (up2);\n    \n     \\path[-] (dp3) edge[draw=blue] (up2);\n     \n    \n     \n     \\node (h1) at (7, 4) {};\n     \\node (h2) at (7, -2) {};\n     \n  \n     \\node[circle, draw=green!40] (f1) at (8, 3) {$f_1$};\n     \\node[circle, draw=green!40] (f2) at (8, 1) {$f_2$};\n     \\node[circle, draw=green!40] (f3) at (8, -1) {$f_3$};\n     \n     \\node[circle, draw=green!40] (u1) at (10, 2) {$u_1$};\n     \\node[circle, draw=green!40] (u2) at (10, 0) {$u_2$};\n     \n     \\node[circle, draw=green!40] (fp1) at (12, 3) {$f'_1$};\n     \\node[circle, draw=green!40] (fp2) at (12, 1) {$f'_2$};\n     \\node[circle, draw=green!40] (fp3) at (12, -1) {$f'_3$};\n\n     \\path[-] (h1) edge[draw=black] (h2);\n     \n     \\path[-] (fp1) edge[draw=green] (u1);\n     \\path[-] (fp2) edge[draw=green] (u2);\n     \\path[-] (u2) edge[draw=green] (u1);\n     \n     \\path[-] (fp2) edge[draw=green] (u1);\n     \\path[-] (fp3) edge[draw=green] (u2);\n     \\path[-] (f1) edge[draw=green] (u1);\n     \\path[-] (f2) edge[draw=green] (u2);\n     \n     \\path[-] (f2) edge[draw=green] (u1);\n     \\path[-] (f3) edge[draw=green] (u2);\n     \n     \\path[-] (fp1) edge[draw=green] (f1);\n    \n    \n    \n    \n    \n    \n    \n     \\path[-] (fp3) edge[draw=green] (f3);\n     \n    \n    \\end{tikzpicture}\n    \n    \\caption{Absorbers $\\xi_5(u)$ (left) and $\\xi'_5(u_1, u_2)$ (right) with $r=5$, where $u, u_1, u_2$ are the green vertices, the green vertices are inside the $A'$-neighborhood of $u$ (or $u_1, u_2$ respectively) and the blue vertices are vertices chosen from $A'$.}\n    \\label{fig:absorber_odd}\n\\end{figure}\n\n\nSubsequently, we absorb $C$ using independent copies of $\\xi_r(\\cdot)$ such that the parity of the remaining vertices in the almost cliques is even. Then, as above, we glue the absorbers together by GO and are left with sets $A''$ and $B''$ (hence, $A'$ without the glued structures) and the path-like subgraph $P'_A$ of size at most $25r \\alpha n$, hence $\\abs{A''} \\geq (\\tfrac 12 - 30 r \\alpha n)$. As each vertex of $C$ has degree at least $\\tfrac{\\alpha}{100} n$ into $A''$ and $B''$, we can absorb one vertex such that in the end $A''$ and $B''$ contain an even number of vertices. Furthermore, each vertex in $A''$ is connected to at least $(\\tfrac 12 - 40 r \\alpha)n$ vertices in $A''$ and the same applies to $B''$. \n\nAgain, as in the even case, we need to make sure that $\\abs{A''} \\equiv 0 \\pmod{2r}$, as we want to close the cycle by blocks of subsequently fully connected layers of sizes $\\tfrac{r+1}{2}$ where the second block misses one matching of size $r+1$. If the divisibility condition holds, set $A''' = A''$ and proceed. Otherwise, if there is $0 < i < 2r$ such that $\\abs{A''} \\equiv i \\pmod{2r}$, $i$ has to be even as the parity of $A''$ guarantees. Select $\\tfrac i2$ pairs vertices $(a_{11}, a_{12}), \\ldots,(a_{i\/2, 1} a_{i\/2, 2}) \\in A'' \\times A''$ and absorb them using disjoint instances $\\xi'_r(a_{11}, a_{12}), \\ldots, \\xi'_r(a_{i\/2, 1} a_{i\/2, 2})$ with $D = A''$. As each absorber consumes $r + 3$ vertices, the divisibility condition now holds. Finally, as in the even case, glue the absorbed parts together with GO which does not change the divisibility by $r+1$. Thus, we are left with a set $A'''$ which consists of the vertices of $A''$ without the absorbed vertices and the gluing structures. Analogously, the same applies for $B'''$. Now, as in the even case, the result directly follows from Lemma~\\ref{lem:blowup} and the following claim (\\ref{stepd:cover} of Section \\ref{sec:o_extremal1}).\n\n\\begin{claim}\nThe constructed subgraph is $r$-connected and $r$-regular.\n\\end{claim}\n\\begin{claimproof}\nAs in the even case, $r$-regularity as well as $r$-connectivity on the $\\tfrac{r+1}{2}$-blow-up of the path part is obvious.\nThe first type of bridge (build with Claim~\\ref{lemma_matching_edges_odd}, see Figure~\\ref{fig:bridge_graph_h_odd} on the left) does not harm connectivity as before. In the second type of bridge (build with Claim~\\ref{cor_matching_edges_odd}, see Figure~\\ref{fig:bridge_graph_h_odd} on the right) only the special vertices $x_a$ and $x_b$ need our attention. But as they are of degree $r$ and connected to $(\\tfrac{r-1}{2}$ vertices on both sides of the $K_{(r+1)\/2), (r+1)\/2}$, isolating a part of the graph is not possible either. The absorbing structures clearly sustain the connectivity property.\n\\end{claimproof}\nThis finishes the proof of the first extremal case.\n\\end{proof}\n\n\\@ifstar{\\origsection*}{\\mysection}{Extremal Case II}\n\\label{sec:extremal2}\nIn this section we deal with the second extremal case and follow~\\ref{stepe:find}--\\ref{stepe:cover} as outlined in Section~\\ref{sec:o_extremal2}.\nWe start by proving the auxiliary lemma for finding stars.\n\n\\begin{proof}[Proof of Lemma~\\ref{lem:stars}]\nLet $\\alpha = \\tfrac{1}{32s^2(s+1)}$.\nAssume we have already found $0 \\le t < 2m$ copies of $K_{1,s}$ and let $V'$ be the remaining vertices.\nThen, by the maximum degree condition in $G$,\n\\[e(G[V']) \\ge \\tfrac 12 n (m+s-1) - t (s+1) 4 s \\alpha n \\,.\\]\nIf $m \\ge s+1$ this is at least $\\tfrac 14 n (m+s-1) \\ge \\tfrac 12 ns$ and gives a vertex of degree at least $s$ in $G[V']$.\nOn the other hand, if $m \\le s$ the above is at least $(\\tfrac 12 s - \\tfrac 14) n > \\tfrac{s-1}{2}n$ and again this gives a vertex of degree at least $s$ in $G[V']$.\n\\end{proof}\n\n\\begin{proof}[Proof of Extremal Case II]\nLet $r \\ge 2$ and $s = \\lceil \\tfrac{r}{2} \\rceil \\ge 1$.\nLet $\\eps>0$ be given by Lemma~\\ref{lem:blowup} on input $\\tfrac 12$, $\\tfrac 12$, and $r$.\nWe obtain $\\alpha>0$ from Lemma~\\ref{lem:stars} and additionally assume that $40 s^2 \\alpha < \\eps$.\nThen let $G$ be an $n$-vertex graph with minimum degree $\\delta(G) \\ge \\tfrac{n+r-2}{2}$ and $nr \\equiv 0 \\pmod 2$.\nFurther assume, that there is a partition of $V(G)$ into $A$ and $B$ of size $|A|+m=\\tfrac 12 n=|B|-m$, where $0 \\le m \\le \\alpha n$ such that between these sets we have minimum degree $\\alpha n$ and all but at most $\\alpha n$ vertices from $A$ (or $B$) have degree $|B|-\\alpha n$ (or $|A|-\\alpha n$) into $B$ (or $A$).\n\n\\noindent \\textbf{\\ref{stepe:find}.}\nNote $\\delta(G[B]) \\ge m+s-1$.\nLet $B' \\subseteq B$ be the vertices of degree at most $2 s \\alpha n$ in $G[B]$ and let $m'=|B \\setminus B'|$.\nIf $m'<m$, then $\\delta(G[B']) \\ge (m-m')+s-1$ and we apply Lemma~\\ref{lem:stars} to find $2(m-m')$ copies of $K_{1,s}$.\nBy choice of $B'$ each vertex from these copies of $K_{1,s}$ has degree at least $|A|-2s\\alpha n$ into $A$.\nLet $W$ be the union of the vertices from these copies of $K_{1,s}$\n\nAfterwards, for $i=1,\\dots,\\min\\{m', m\\}$ we can pick a vertex $x_i \\in B \\setminus (B' \\cup \\{x_1,\\dots,x_{i-1} \\})$ and neighbours $y_{i,1},\\dots,y_{i,r}$ of $x_i$ from $A \\setminus (W \\cup \\bigcup_{j=1}^{i-1} \\{y_{j,1},\\dots,y_{j,r} \\}$ such that $y_{i,1},\\dots,y_{i,r}$ have degree at least $|A|-\\alpha n$  into $A$.\nFor some $0 \\le m' \\le m$ we have obtained $m'$ copies of  $K_{1,r}$ and $2(m-m')$ copies of $K_{1,s}$ such that all vertices in the copies of $K_{1,s}$ and the leaves in the copies of $K_{1,r}$ have degree at least $|A|-2s\\alpha n$ into $A$.\n\n\\noindent \\textbf{\\ref{stepe:absorb}.}\nWe will now iteratively absorb these copies of $K_{1,s}$ and $K_{1,r}$ into an $r$-regular path-structure.\nLet $W$ be the set of vertices in the union of these copies.\nWe start with $r$ vertices $\\mathbf{v}=(v_1,\\dots,v_s)$ from $A \\setminus W$ that have degree at least $|A|-\\alpha n$ into $A$ and $s$ vertices $\\mathbf{u}_0=(u_{0,1},\\dots,u_{0,s})$ from $B \\setminus W$ that have degree at least $|B|-\\alpha n$ into $B$ such that $v_ju_{0,j'}$ is an edge for $1 \\le j, j' \\le s$.\nWe add the vertices in $\\mathbf{v}$ and $\\mathbf{u}_0$ to $W$.\nNow for some $i=0,\\dots,m$ let $W$ be the vertices used for the structure and in the remaining copies of $K_{1,s}$ and $K_{1,s}$ with $|W| \\le 6 s i + 2s$ and let $\\mathbf{u}_i=(u_{i,1},\\dots,u_{i,r})$ be one end of the structure in $B$ such that $u_{i,j}$ has at least $|A|-2s\\alpha n$ neighbours in $A$ for $j=1,\\dots,s$.\n\nIf there are two copies of $K_{1,s}$ left then by alternating between $A$ and $B$ we find the structure shown in Figure~\\ref{fig:absorbK1s} where the vertices on the left are $\\mathbf{u}_i$ and the vertices on the right we denote by $\\mathbf{u}_{i+1}=(u_{i+1,1},\\dots,u_{i+1,s})$.\nNote that $u_{i+1,1},\\dots,u_{i+1,s}$ are exactly the leaves of one of the $K_{1,s}$ and, therefore, have at least degree $|A|-2s\\alpha n$ into $A$.\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8]\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b11) at (-6, 0) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b12) at (-6, -1.5) {};\n\n     \\node[circle, draw=blue, fill=blue] (a11) at (-4, 0) {};\n     \\node[circle, draw=blue, fill=blue] (a12) at (-4, -1.5) {};\n     \n     \\node[circle, draw=green, fill=green!60] (b1) at (-3, -0.75) {};\n     \n     \\node[circle, draw=black!60!green, fill=black!60!green] (b21) at (-2, 0) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b22) at (-2, -1.5) {};\n\n     \\node[circle, draw=blue, fill=blue] (a21) at (-0, 0) {};\n     \\node[circle, draw=blue, fill=blue] (a22) at (-0, -1.5) {};\n     \n     \\node[circle, draw=green, fill=green!60] (b2) at (1, -0.75) {};\n     \n     \\node[circle, draw=black!60!green, fill=black!60!green] (b31) at (2, 0) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b32) at (2, -1.5) {};\n\n\n     \n     \\path[-] (b11) edge[draw=black] (a11);\n     \\path[-] (b11) edge[draw=black] (a12);\n     \\path[-] (b12) edge[draw=black] (a11);\n     \\path[-] (b12) edge[draw=black] (a12);\n    \n     \\path[-] (a11) edge[draw=black] (b21);\n     \\path[-] (a11) edge[draw=black] (b1);\n     \\path[-] (a12) edge[draw=black] (b22);\n     \\path[-] (a12) edge[draw=black] (b1);\n     \n     \\path[-] (b1) edge[draw=green!60] (b21);\n     \\path[-] (b1) edge[draw=green!60] (b22);\n     \n     \\path[-] (b21) edge[draw=black] (a21);\n     \\path[-] (b21) edge[draw=black] (a22);\n     \\path[-] (b22) edge[draw=black] (a21);\n     \\path[-] (b22) edge[draw=black] (a22);\n    \n     \\path[-] (a21) edge[draw=black] (b31);\n     \\path[-] (a21) edge[draw=black] (b2);\n     \\path[-] (a22) edge[draw=black] (b32);\n     \\path[-] (a22) edge[draw=black] (b2);\n     \n     \\path[-] (b2) edge[draw=green!60] (b31);\n     \\path[-] (b2) edge[draw=green!60] (b32);\n\n   \n   \n    \n   \n    \\end{tikzpicture}\n    \\qquad \n    \\begin{tikzpicture}[scale=0.8]\n    \\node[circle, draw=black!60!green, fill=black!60!green] (b11) at (-6, 1) {};\n    \\node[circle, draw=black!60!green, fill=black!60!green] (b12) at (-6, -1) {};\n    \\node[circle, draw=black!60!green, fill=black!60!green] (b13) at (-6, -3) {};\n\n     \\node[circle, draw=blue, fill=blue] (a11) at (-4, 1) {};\n     \\node[circle, draw=blue, fill=blue] (a12) at (-4, -1) {};\n     \\node[circle, draw=blue, fill=blue] (a13) at (-4, -3) {};\n     \n     \\node[circle, draw=green, fill=green!60] (b1) at (-3, -1) {};\n     \n     \\node[circle, draw=black!60!green, fill=black!60!green] (b21) at (-2, 1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b22) at (-2, -1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b23) at (-2, -3) {};\n\n     \\node[circle, draw=blue, fill=blue] (a21) at (-0, 1) {};\n     \\node[circle, draw=blue, fill=blue] (a22) at (-0, -1) {};\n     \\node[circle, draw=blue, fill=blue] (a23) at (-0, -3) {};\n     \n     \\node[circle, draw=green, fill=green!60] (b2) at (1, -1) {};\n     \n     \\node[circle, draw=black!60!green, fill=black!60!green] (b31) at (2, 1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b32) at (2, -1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b33) at (2, -3) {};\n\n\n    \n     \\path[-] (b11) edge[draw=black] (a12);\n     \\path[-] (b11) edge[draw=black] (a13);\n     \\path[-] (b12) edge[draw=black] (a11);\n    \n     \\path[-] (b12) edge[draw=black] (a13);\n     \\path[-] (b13) edge[draw=black] (a11);\n     \\path[-] (b13) edge[draw=black] (a12);\n    \n    \n     \\path[-] (a11) edge[draw=black] (b22);\n     \\path[-] (a11) edge[draw=black] (b1);\n     \\path[-] (a11) edge[draw=black, bend left = 15] (b23);\n     \n     \\path[-] (a12) edge[draw=black] (b21);\n     \\path[-] (a12) edge[draw=black] (b1);\n     \\path[-] (a12) edge[draw=black] (b23);\n     \n     \\path[-] (a13) edge[draw=black, bend left = 15] (b21);\n     \\path[-] (a13) edge[draw=black] (b1);\n     \\path[-] (a13) edge[draw=black] (b22);\n     \n     \\path[-] (b1) edge[draw=green!60] (b21);\n     \\path[-] (b1) edge[draw=green!60] (b22);\n     \n    \n     \\path[-] (b21) edge[draw=black] (a22);\n     \\path[-] (b21) edge[draw=black] (a23);\n     \\path[-] (b22) edge[draw=black] (a21);\n    \n     \\path[-] (b22) edge[draw=black] (a23);\n     \\path[-] (b23) edge[draw=black] (a21);\n     \\path[-] (b23) edge[draw=black] (a22);\n     \\path[-] (b23) edge[draw=black] (a23);\n    \n     \\path[-] (a21) edge[draw=black] (b32);\n     \\path[-] (a21) edge[draw=black] (b2);\n     \\path[-] (a21) edge[draw=black, bend left = 15] (b33);\n    \n     \\path[-] (a22) edge[draw=black] (b31);\n     \\path[-] (a22) edge[draw=black] (b2);\n     \\path[-] (a22) edge[draw=black] (b33);\n     \n     \\path[-] (a23) edge[draw=black, bend left = 15] (b31);\n     \\path[-] (a23) edge[draw=black] (b32);\n    \n     \\path[-] (b2) edge[draw=green!60] (b31);\n     \\path[-] (b2) edge[draw=green!60] (b32);\n     \\path[-] (b2) edge[draw=green!60] (b33);\n     \\end{tikzpicture}\n    \n    \\caption{Including two copies of $K_{1,s}$ into an $r$-regular path structure in the even ($r=4$) and odd case ($r=5$) with blue vertices in $A$ and green vertices in $B$.}\n    \\label{fig:absorbK1s}\n\\end{figure}\n\nOtherwise, we pick a copy of $K_{1,r}$ and find the structure shown in Figure~\\ref{fig:absorbK1r} (this is an $\\tfrac r2$-blow-up of a path), where the vertices on the left are $\\mathbf{u}_i$ and the vertices on the right we denote by $\\mathbf{u}_{i+1}=(u_{i+1,1},\\dots,u_{i+1,s})$.\nWe do not need any assumption on the degree of the centre vertex of the $K_{1,r}$ as we do not need any additional edges containing it.\nAgain $u_{i+1,1},\\dots,u_{i+1,r}$ have at least degree $|A|- 2s\\alpha n$ into $A$.\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8]\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b11) at (-6, 0) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b12) at (-6, -1.5) {};\n\n     \\node[circle, draw=blue, fill=blue] (a11) at (-4, 0) {};\n     \\node[circle, draw=blue, fill=blue] (a12) at (-4, -1.5) {};\n     \n     \\node[circle, draw=black!60!green, fill=black!60!green] (b21) at (-2, 0) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b22) at (-2, -1.5) {};\n\n     \\node[circle, draw=green!60, fill=green!60] (b1) at (-0, 0) {};\n     \\node[circle, draw=blue, fill=blue] (a22) at (-0, -1.5) {};\n\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b31) at (2, 0) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b32) at (2, -1.5) {};\n\n\n     \n     \\path[-] (b11) edge[draw=black] (a11);\n     \\path[-] (b11) edge[draw=black] (a12);\n     \\path[-] (b12) edge[draw=black] (a11);\n     \\path[-] (b12) edge[draw=black] (a12);\n    \n     \\path[-] (a11) edge[draw=black] (b21);\n     \\path[-] (a11) edge[draw=black] (b22);\n     \\path[-] (a12) edge[draw=black] (b22);\n     \\path[-] (a12) edge[draw=black] (b21);\n\n     \\path[-] (b21) edge[draw=green!60] (b1);\n     \\path[-] (b21) edge[draw=black] (a22);\n     \\path[-] (b22) edge[draw=green!60] (b1);\n     \\path[-] (b22) edge[draw=black] (a22);\n    \n     \\path[-] (b1) edge[draw=green!60] (b31);\n     \\path[-] (a22) edge[draw=black] (b31);\n     \\path[-] (b1) edge[draw=green!60] (b32);\n     \\path[-] (a22) edge[draw=black] (b32);\n     \n    \\end{tikzpicture}\n    \\qquad \n    \\begin{tikzpicture}[scale=0.8]\n    \\node[circle, draw=black!60!green, fill=black!60!green] (b11) at (-6, 1) {};\n    \\node[circle, draw=black!60!green, fill=black!60!green] (b12) at (-6, -1) {};\n    \\node[circle, draw=black!60!green, fill=black!60!green] (b13) at (-6, -3) {};\n\n     \\node[circle, draw=blue, fill=blue] (a11) at (-4, 1) {};\n     \\node[circle, draw=blue, fill=blue] (a12) at (-4, -1) {};\n     \\node[circle, draw=blue, fill=blue] (a13) at (-4, -3) {};\n\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b21) at (-2, 1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b22) at (-2, -1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b23) at (-2, -3) {};\n\n     \\node[circle, draw=green!60, fill=green!60] (b1) at (-0, 1) {};\n     \\node[circle, draw=blue, fill=blue] (a22) at (-0, -1) {};\n     \\node[circle, draw=blue, fill=blue] (a23) at (-0, -3) {};\n     \n\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b31) at (2, 1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b32) at (2, -1) {};\n     \\node[circle, draw=black!60!green, fill=black!60!green] (b33) at (2, -3) {};\n\n\n    \n     \\path[-] (b11) edge[draw=black] (a12);\n     \\path[-] (b11) edge[draw=black] (a13);\n     \\path[-] (b12) edge[draw=black] (a11);\n    \n     \\path[-] (b12) edge[draw=black] (a13);\n     \\path[-] (b13) edge[draw=black] (a11);\n     \\path[-] (b13) edge[draw=black] (a12);\n    \n    \n     \\path[-] (a11) edge[draw=black] (b22);\n     \\path[-] (a11) edge[draw=black] (b21);\n     \\path[-] (a11) edge[draw=black] (b23);\n     \n     \\path[-] (a12) edge[draw=black] (b21);\n     \\path[-] (a12) edge[draw=black] (b22);\n     \\path[-] (a12) edge[draw=black] (b23);\n     \n     \\path[-] (a13) edge[draw=black] (b21);\n     \\path[-] (a13) edge[draw=black] (b22);\n     \\path[-] (a13) edge[draw=black] (b22);\n     \n    \n     \\path[-] (b21) edge[draw=black] (a22);\n     \\path[-] (b21) edge[draw=black] (a23);\n     \\path[-] (b22) edge[draw=green!60] (b1);\n    \n     \\path[-] (b22) edge[draw=black] (a23);\n     \\path[-] (b23) edge[draw=green!60] (b1);\n     \\path[-] (b23) edge[draw=black] (a22);\n    \n    \n     \\path[-] (b1) edge[draw=green!60] (b32);\n     \\path[-] (b1) edge[draw=green!60] (b31);\n     \\path[-] (b1) edge[draw=green!60] (b33);\n    \n     \\path[-] (a22) edge[draw=black] (b31);\n     \\path[-] (a22) edge[draw=black] (b32);\n     \\path[-] (a22) edge[draw=black] (b33);\n     \n     \\path[-] (a23) edge[draw=black] (b31);\n     \\path[-] (a23) edge[draw=black] (b32);\n     \\path[-] (a23) edge[draw=black] (b33);\n\n     \\end{tikzpicture}\n    \n    \\caption{Including one copy of $K_{1,r}$ into an $r$-regular path structure in the even ($r=4$) and odd case ($r=5$) with blue vertices in $A$ and green vertices in $B$.}\n    \\label{fig:absorbK1r}\n\\end{figure}\n\nWe add all new vertices in the structure to $W$. We can repeat this until all copies of $K_{1,s}$ and $K_{1,r}$ are covered, because in each step $W$ will increase by at most $6r$ vertices and all vertices have large enough neighbourhoods.\nWe let $\\mathbf{u}'=(u_{m,1},\\dots,u_{m,s})$ be the final end of this construction.\nNow let $A_1=A \\setminus W$ and $B_2=B \\setminus W$ and note that $|A_1|=|B_1|$ because our construction uses $2m$ vertices more from $B$ than from $A$.\n\n\\noindent \\textbf{\\ref{stepe:absorb2}.}\nFor the next step let $A' \\subseteq A_1$ be the vertices of degree at most $|A_1|-5 s \\alpha n$ into $B_1$ and  $B' \\subseteq B_1$ be the vertices of degree at most $|A_1|-5 s \\alpha n$ into $A_1$.\nNote that $|A'|,|B'| \\le \\alpha n$, because we removed at most $4 s \\alpha n$ vertices from each of $A$ and $B$ to get $A_1$ and $B_1$.\nBy using an $\\tfrac r2$-blow-up of a path (similar as in Figure~\\ref{fig:absorbK1r}) we cover the vertices in $A'$ and $B'$ with our $r$-regular path structure covering in total at most $12 s \\alpha n$ additional vertices.\nFor this we iteratively extend from $\\mathbf{u}'$.\nLet $W$ be the vertices used for covering $A'$ and $B'$ and let $\\mathbf{u}=(u_1,\\dots,u_s)$ be the last vertices of this construction in $B$ and note that we can assume that they have degree at least $|A|-20 s \\alpha n$ into $A$.\nWe let $A_2=(A_1 \\setminus W)$ and $B_2=(B_1 \\setminus W)$ and note that $|A_2|=|B_2|$.\n\n\\noindent \\textbf{\\ref{stepe:cover}.}\nWe have that every vertex from $A_2$ (or $B_2$) has degree at least $|A_2|-20 s \\alpha n$ into $B_2$ (or into $A_2$).\nThen it is easy to see that $(A_2,B_2)$ is $(\\eps,\\tfrac 12)$-super-regular as $40 s \\alpha < \\eps$.\nMoreover, the vertices in $\\mathbf{v}$ have $|B_2| - 20s^2\\alpha n \\ge \\tfrac 12 |B_2|$ common neighbours in $B_2$ and similarly for $\\mathbf{u}$ with $A_2$.\nWe apply Lemma~\\ref{lem:blowup} to cover the remaining vertices of $A_2$ and $B_2$ with the $\\tfrac r2$-blow-up of a path such that the ends connect to $\\mathbf{v}$ and $\\mathbf{u}$.\nThis completes the construction of a spanning $r$-regular structure in $G$.\nTo see that it is also $r$-connected it suffices to note that we have the $\\tfrac r2$-blow-up of a path except for some parts replaced by the graphs from Figure~\\ref{fig:absorbK1s}, which do not harm this property.\n\\end{proof}\n\n\\@ifstar{\\origsection*}{\\mysection}{Non-Extremal Case}\n\\label{sec:non-extremal}\n\nIn this section we deal with the case that $G$ is not $\\alpha$-extremal.\nRecall, that the assumption implies that for any two sets $A,B \\subseteq V(G)$ of size $(\\tfrac12 - \\alpha)n \\le |A|,|B| \\le \\tfrac n2$ we have $d(A,B) \\ge \\alpha$.\nWe will follow~\\ref{step:regularise}--\\ref{step:spanning} as outlined in Section~\\ref{sec:o_non-extremal}.\n\n\\begin{proof}[Proof of Non-Extremal Case]\nGiven $r \\ge 3$ and $0<\\alpha<\\tfrac{1}{32}$ we choose constants such that\n\\begin{align*}\n    \\varepsilon \\ll \\nu \\ll d \\ll \\beta \\ll \\alpha\\,,\n\\end{align*}\nwhere, in particular,\n\\begin{align*}\n    \\eps \\le \\nu, \\qquad \\nu \\le d^{s+1}, \\qquad 10 s d \\le \\beta, \\qquad 500 s \\beta \\le \\alpha \\,\n\\end{align*}\nand $2 \\eps$ is small enough for Lemma~\\ref{lem:blowup} with input $\\tfrac d2$, $\\tfrac 14 d^s$, and $r$.\nLet $M$ be given by Lemma~\\ref{lem:regularity} on input $\\eps$ and let $s=\\lceil \\tfrac r2 \\rceil$.\n\n\\smallskip\n\\noindent \\textbf{\\ref{step:regularise}.}\nLet $G$ be an $n$-vertex graph with minimum degree $\\delta(G) \\ge (\\tfrac 12-\\beta)n$.\nFrom Lemma~\\ref{lem:regularity} we get a partition of the vertex set $V(G)$ into $\\ell+1 \\le M$ clusters $V_0,\\dots,V_\\ell$ of size $L$ and a subgraph $G' \\subseteq G$ such that~\\ref{reg:size}--\\ref{reg:reg} hold.\nWe denote by $R$ the graph on vertex set $[\\ell]$ with edges $ij$ if and only if the pair $(V_i,V_j)$ is $(\\eps,d)$-regular.\nIn $R$ we have minimum degree $\\delta(R) \\ge (\\tfrac 12-\\beta-2d)\\ell$, because otherwise with~\\ref{reg:ind} there would be a vertex with degree in $G'$ at most $(\\tfrac 12-\\beta-2d) \\ell \\cdot \\tfrac{n}{\\ell} + \\eps n < (\\tfrac 12-\\beta) n - (d+\\eps)n$ contradicting~\\ref{reg:deg}.\nSimilarly, we can deduce that $R$ is not $\\tfrac \\alpha 2$-extremal.\nOtherwise, there would be two sets of vertices $\\mathcal{A}$, $\\mathcal{B}$ in $R$ such that $(\\tfrac 12-\\tfrac \\alpha2 )\\ell \\le |\\mathcal{A}|,|\\mathcal{B}| \\le \\tfrac 12 \\ell$ and $d(\\mathcal{A},\\mathcal{B})< \\tfrac \\alpha2$.\nThen $A=\\bigcup_{i \\in \\mathcal{A}} V_i$ and $B=\\bigcup_{i \\in \\mathcal{B}} V_i$ both have size at most $\\tfrac \\ell 2 \\cdot \\tfrac{n}{\\ell} = \\tfrac n2$ and at least $(\\tfrac 12- \\tfrac \\alpha 2) \\ell \\cdot (1-\\eps) \\tfrac{n}{\\ell} \\ge (\\tfrac 12-\\alpha)n$ and we have\n\\[ d(A,B) = \\frac{e(A,B)}{|A||B|} \\le \\frac{\\tfrac \\alpha 2 \\cdot (\\tfrac n \\ell)^2 \\cdot |\\mathcal{A}||\\mathcal{B}| + |A| (d+\\eps) n}{|A| |B|} \\le \\frac{\\tfrac \\alpha 2}{(1-\\eps)^2} + \\frac{(d+\\eps)}{\\tfrac 12-\\alpha} \\le \\alpha\\,,\\]\nwhich contradicts our assumption that $G$ is not $\\alpha$-extremal.\nWe will repeatedly use the following fact that holds as $R$ is not $\\tfrac \\alpha2$-extremal.\n\\begin{fact}\n\\label{fact:non-extremal}\n    For any two sets $\\mathcal{A}, \\mathcal{B} \\subseteq V(R)$ of size at least $(\\tfrac 12 - \\tfrac \\alpha 2)\\ell$ there is an edge $AB \\in E(R)$ with $A \\in \\mathcal{A}$ and $B \\in \\mathcal{B}$.\n\\end{fact}\nIn the following we will abuse notation and also treat the clusters as vertices of $R$.\n\n\\smallskip\n\\noindent \\textbf{\\ref{step:matching}.}\nNext let $M$ be a largest matching in $R$ and $M_0 \\subseteq [\\ell]$ be the clusters not covered by $M$.\nNaturally, $M_0$ is an independent set in $R$ and if there are at least two vertices $u$ and $v$ in $M_0$ then no neighbour of $u$ is connected to a neighbour of $v$ by an edge of $M$.\nTherefore, $2|M| \\ge \\deg_R(u)+\\deg_R(v)$ and $|M | \\ge (\\tfrac 12-\\beta-2d) \\ell$.\nWe let $\\ell'=|M|$ and denote the regular pairs corresponding to edges of $M$ by $(X_i,Y_i)$ for $i=1,\\dots,\\ell'$.\n\n\\smallskip\n\\noindent \\textbf{\\ref{step:connect}.}\nFor any clusters $Z$ and $W$ we call an $s$-tuple $\\mathbf{z}=(z_1,\\dots, z_s)$ from $Z$ \\emph{well-connected into $W$} if the vertices $z_1,\\dots,z_s$ have at least $\\tfrac 12 d^s L$ common neighbours in $W$.\nNow fix any $i \\in [\\ell']$.\nWe want to connect $Y_i$ and $X_{i+1}$ by the $\\tfrac r2$-blow-up of a path.\nFor this we consider the neighbours $\\mathcal{W}$ and $\\mathcal{Z}$ in $R$ of $X_{i+1}$ and $Y_i$, respectively.\nIt follows from the minimum degree in $R$ that $\\mathcal{W}$ and $\\mathcal{Z}$ have size at least $(\\tfrac 12 - \\tfrac \\alpha 2)\\ell$.\nBy Fact~\\ref{fact:non-extremal} there is $W \\in \\mathcal{W}$ and $Z \\in \\mathcal{Z}$ such that $WZ$ is an edge in $R$.\n\nAll but at most $2 s \\eps L^s$ $s$-tuples $\\mathbf{x}=(x_1,\\dots, x_s)$ from $X_{i+1}$ are well connected into $W$ and $Y_{i+1}$.\nThe same holds for tuples from $Y_i$, $W$, and $Z$ with respect to the neighbouring clusters.\nWe fix tuples $\\mathbf{x}=(x_1,\\dots, x_s)$, $\\mathbf{w}=(w_1,\\dots, w_s)$, $\\mathbf{z}=(z_1,\\dots, z_s)$, and $\\mathbf{y}=(y_1,\\dots, y_s)$ from $X$, $W$, $Z$, and $Y$, respectively, such that $\\mathbf{x}\\mathbf{w}\\mathbf{z}\\mathbf{y}$ gives the $\\tfrac r2$-blow up of a path on $4$ vertices.\nWe denote this path by $P_i$ and remove any internal vertices (those in $\\mathbf{w}$ and $\\mathbf{z}$) from the clusters.\nWe can repeat this for all $i$, because we need only $4 s \\ell'$ vertices in total.\n\n\\smallskip\n\\noindent \\textbf{\\ref{step:superreg}.}\nTo make the matching edges super-regular, we let $i \\in [\\ell']$ and apply Lemma~\\ref{lem:superreg} to the pair $(Y_i,X_i)$.\nAfter removing a few additional vertices we arrive at sets $Y_i$ and $X_i$ such that $|Y_i|=|X_i| = L' \\ge (1-2\\eps)L$, where $L' \\equiv 0 \\pmod{s}$, and the pair $(Y_i,X_i)$ is $(2 \\eps, d-2\\eps)$-super-regular.\nNote that we can do this such that the end-tuple $\\mathbf{x}$ of $P_{i-1}$ is contained in $X_i$ and the end-tuple $\\mathbf{y}$ of $P_{i}$ is contained in $Y_i$.\nWe add the vertices removed during this procedure and also the vertices that belong to clusters of $M_0$ to $V_0$ and note that $|V_0| \\le \\eps n + \\ell' 4 \\eps L + \\beta n + 2dn \\le 2 \\beta n$.\n\n\\smallskip\n\\noindent \\textbf{\\ref{step:absorb}. Setup.}\nWe want to absorb $V_0$ by extending the paths $P_i$.\nAfter each extension we need to maintain the location of the end-tuples and also ensure that they are well-connected.\nDuring the procedure we will have to deal with sets of already covered vertices.\nFor this let $W_0$ be a set of size at most $18 s^2 \\nu n$ and let $W$ be a set of size at most $20 s \\beta n$.\nThese will be the sets of vertices that we already used.\nThere are at most $18 s^2 \\nu n \/ (\\tfrac 14 d^s \\tfrac n\\ell) = 72 s^2 \\nu d^{-s}  \\ell \\le 8 s \\beta \\ell$ clusters that intersect $W_0$ in at least $\\tfrac 14 d^s \\tfrac n\\ell$ vertices and at most $20 s \\beta n \/ (\\tfrac 12 \\cdot \\tfrac n\\ell) \\le 40 s \\beta \\ell$ clusters that intersect $W$ in at least $\\tfrac 12 \\cdot \\tfrac n\\ell$ vertices.\nWe denote by $H$ the set of all clusters that do not have this property.\n\nNow consider a vertex $v \\in V_0$.\nThere are at most $(\\tfrac 12 + 50 s \\beta) \\ell$ clusters that intersect $N_G(v) \\setminus (W \\cup W_0)$ in less than $d \\tfrac n \\ell$ vertices.\nTherefore, there are at least $(\\tfrac 12 - 100 s \\beta) \\ell$ clusters in $H$ that intersect $N_G(v) \\setminus (W \\cup W_0)$ in at least $d \\tfrac n \\ell$ vertices.\nWe denote this set of clusters by $H(v)$.\nSimilarly, let $H_M(v)$ be the clusters form $H$, which share an edge of $M$ with another cluster from $H(v)$ and note that we have the same lower bound as $M$ is a matching.\nSumming up we have $|H| \\ge (1-50 s \\beta)\\ell$ and $|H(v)|,|H_M(v)| \\ge (\\tfrac 12 - 100 s \\beta) \\ell$ for all $v \\in V_0$.\nNote that $H(v)$ and $H_M(v)$ are large enough for Fact~\\ref{fact:non-extremal}.\n\n\\smallskip\n\\textbf{Covering $2s$ vertices.}\nWe let $W=\\emptyset$ and $W_0$ be all internal vertices (not in end-tuples) of the paths $P_1,\\dots,P_\\ell$.\nWe pick any $2s$ vertices $v_1,\\dots,v_{2s}$ from $V_0 \\setminus (W \\cup W_0)$ and our goal is to embed them, such that in any pair $(X_i,Y_i)$ we still have $|X_i|=|Y_i| \\equiv 0 \\pmod s$.\nLet $i_1$ be such that $X_{i_1} \\in H(v_1)$ and $Y_{i_1} \\in H$.\nAs the end-tuple of $P_{i_1}$ in $X_{i_1}$ is well-connected into $Y_{i_1}$ and $(N_G(v)\\cap X_{i_1}) \\setminus (W \\cup W_0)$ is of size at least $d \\tfrac n \\ell$, we can greedily pick tuples $\\mathbf{x}_1$, $\\mathbf{x}_2$ in $X_{i_1}$ and $\\mathbf{y}_1$, $\\mathbf{y}_2$ in $Y_{i_1}$ with the exception that $\\mathbf{y}_1$ contains $v_1$ and such that $\\mathbf{x}_1\\mathbf{y}_1\\mathbf{x}_2\\mathbf{y}_2P_{i_1}$ gives the $\\tfrac r2$-blow-up of a path and $\\mathbf{x_1}$ is well-connected into $Y_{i_1}$.\nNow we remove the internal vertices (those in $\\mathbf{x_2}$, $\\mathbf{y_1}$, $\\mathbf{y_2}$ and the end-tuple $\\mathbf{x}$ of $P_{i_1}$) from the clusters and add them to $W_0$ (this adds $4s$ vertices; see Figure~\\ref{fig:Step51}).\nWe note that $|Y_{i_1}|-1=|X_{i_1}|\\equiv 0 \\pmod s$ and let $P_{i_1}$ be the longer path.\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8, every node\/.style={inner sep=0.2,outer sep=0.2}]\n\n    \\node[circle, draw=red, fill = red, minimum width = 6, font=\\tiny] (v1) at (0,0) {};\n    \\node[] at (0.5, 0) {$v_1$};\n    \\node[circle, draw=orange, fill=orange, font=\\tiny, minimum width = 6] (x11) at (-2.7,1) {};\n    \\node[circle, draw=orange, fill=orange, font=\\tiny, minimum width = 6] (x12) at (-2.7,1.5) {};\n    \\node[] at (-3.2, 1.25) {$\\mathbf{x_1}$};\n    \n    \\node[circle, draw=black, fill=black, minimum width = 6, font=\\tiny] (x21) at (-2.25,2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (x22) at (-2.25,2.75) {};\n    \\node[] at (-2.7, 2.5) {$\\mathbf{x_2}$};\n    \n    \\node[circle, draw=black, fill=black, minimum width = 6, font=\\tiny] (y1) at (1,1) {};\n    \\node[] at (1.5, 1) {$y_1$};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (y21) at (1,2.5) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (y22) at (1,3) {};\n    \\node[] at (1.5, 2.75) {$\\mathbf{y_2}$};\n\n    \n    \\node[circle, draw=orange, fill=orange, font=\\tiny, minimum width = 6] (x1) at (-2.7,3.5) {};\n    \\node[circle, draw=orange, fill=orange, font=\\tiny, minimum width = 6] (x2) at (-2.7,4) {};\n    \\node[] at (-3.2, 3.75) {$\\mathbf{x}$};\n    \n    \\node[ellipse, draw = black, minimum width = 60, minimum height = 100, label = above:{$X_{i_1}$}] (Xi) at (-2.75,2.375) {};\n    \\node[ellipse, draw = black, minimum width = 60, minimum height = 100, label = above:{$Y_{i_1}$}] (Yi) at (1.25,2.375) {};\n    \n    \\path[-] (Xi) edge[draw=blue!20, line width=20pt] (Yi);\n    \n    \\path[-] (v1) edge[draw=red] (x11);\n    \\path[-] (v1) edge[draw=red] (x21);\n    \\path[-] (v1) edge[draw=red] (x12);\n    \\path[-] (v1) edge[draw=red] (x22);\n    \n    \\path[-] (y1) edge[draw=black] (x11);\n    \\path[-] (y1) edge[draw=black] (x21);\n    \\path[-] (y1) edge[draw=black] (x12);\n    \\path[-] (y1) edge[draw=black] (x22);\n    \n    \\path[-] (y21) edge[draw=black] (x11);\n    \\path[-] (y22) edge[draw=black] (x21);\n    \\path[-] (y22) edge[draw=black] (x12);\n    \\path[-] (y21) edge[draw=black] (x22);\n    \n    \\path[-] (y21) edge[draw=black] (x1);\n    \\path[-] (y22) edge[draw=black] (x2);\n    \\path[-] (y21) edge[draw=black] (x2);\n    \\path[-] (y22) edge[draw=black] (x1);\n    \n\n\n    \n     \\end{tikzpicture}\n    \n    \\caption{Absorbing vertex $v_1$ in the case $r=2s=4$ if $\\mathbf{x}$ is the current end of path $P_i$ and $\\mathbf{x_1}$ is the new end.}\n    \\label{fig:Step51}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8, every node\/.style={inner sep=0.2,outer sep=0.2}]\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yi) at (-11.2, 2.5) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj1) at (-7.9, 1.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj2) at (-7.9, 2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj3) at (-7.9, 2.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj4) at (-7.9, 3.25) {};\n    \n    \n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj) at (-4.75,1.875) {};\n    \n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj1) at (-5.2, 1) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj2) at (-5.2, 1.5) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj3) at (-5.2, 2.5) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj4) at (-5.2, 3.5) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj5) at (-5.2, 4) {};\n    \\node[] at (-4.7, 3.75) {$P_{i_j}$};\n    \n    \n    \n    \\node[circle, draw=red, fill = red, minimum width = 6, font=\\tiny] (v1) at (0,0) {};\n    \\node[] at (0.5, 0) {$v_t$};\n    \\node[circle, draw=black, fill=black, font=\\tiny, minimum width = 6] (x11) at (-1.7,1) {};\n    \\node[circle, draw=black, fill=black, font=\\tiny, minimum width = 6] (x12) at (-1.7,1.5) {};\n\n    \\node[circle, draw=black, fill=black, minimum width = 6, font=\\tiny] (x21) at (-1.7,2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (x22) at (-1.7,2.75) {};\n    \n    \\node[circle, draw=black, fill=black, minimum width = 6, font=\\tiny] (x31) at (-1.7,3.5) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (x32) at (-1.7,4) {};\n\n    \n    \\node[circle, draw=orange, fill=orange, minimum width = 6, font=\\tiny] (y11) at (1,1) {};\n    \\node[circle, draw=orange, fill=orange, minimum width = 6, font=\\tiny] (y12) at (1,1.5) {};\n\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (y21) at (1,2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (y22) at (1,2.75) {};\n    \n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (y31) at (1,3.5) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (y32) at (1,4) {};\n    \n    \\node[] at (1.5, 3.75) {$P_{i_t}$};\n\n\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{i_{t}-1}$}] (Yi) at (-11.2,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{i_j}$}] (Yj) at (-5,2.375) {};\n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{i_j}$}] (Xj) at (-7.9,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{i_t}$}] (Xt) at (-1.7,2.375) {};\n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{i_t}$}] (Yt) at (1.2,2.375) {};\n    \n    \\path[-] (Xt) edge[draw=blue!20, line width=20pt] (Yt);\n    \\path[-] (Xt) edge[draw=black!20, line width=20pt] (Yj);\n    \\path[-] (Xj) edge[draw=blue!20, line width=20pt] (Yj);\n    \\path[-] (Xj) edge[draw=black!20, line width=20pt] (Yi);\n    \n    \\path[-] (v1) edge[draw=red] (x11);\n    \\path[-] (v1) edge[draw=red] (x21);\n    \\path[-] (v1) edge[draw=red] (x12);\n    \\path[-] (v1) edge[draw=red] (x22);\n    \n    \\path[-] (y11) edge[draw=black] (x11);\n    \\path[-] (y12) edge[draw=black] (x11);\n    \\path[-] (y11) edge[draw=black] (x12);\n    \\path[-] (y12) edge[draw=black] (x12);\n    \n    \\path[-] (y21) edge[draw=black] (x11);\n    \\path[-] (y21) edge[draw=black] (x12);\n    \\path[-] (y21) edge[draw=black] (x21);\n    \\path[-] (y21) edge[draw=black] (x22);\n    \n    \\path[-] (y22) edge[draw=black] (x31);\n    \\path[-] (y22) edge[draw=black] (x32);\n    \\path[-] (y22) edge[draw=black] (x21);\n    \\path[-] (y22) edge[draw=black] (x22);\n    \n    \\path[-] (yj) edge[draw=black] (x31);\n    \\path[-] (yj) edge[draw=black] (x32);\n    \\path[-] (yj) edge[draw=black] (x21);\n    \\path[-] (yj) edge[draw=black] (x22);\n    \n    \\path[-] (xj1) edge[draw=black] (yj1);\n    \\path[-] (xj1) edge[draw=black] (yj2);\n    \\path[-] (xj1) edge[draw=black] (yj3);\n    \n    \\path[-] (xj2) edge[draw=black] (yj1);\n    \\path[-] (xj2) edge[draw=black] (yj2);\n    \\path[-] (xj2) edge[draw=black] (yj3);\n    \n    \\path[-] (xj3) edge[draw=black] (yj4);\n    \\path[-] (xj3) edge[draw=black] (yj5);\n    \\path[-] (xj3) edge[draw=black] (yj3);\n    \n    \\path[-] (xj4) edge[draw=black] (yj4);\n    \\path[-] (xj4) edge[draw=black] (yj5);\n    \\path[-] (xj4) edge[draw=black] (yj3);\n    \n    \\path[-] (yi) edge[draw=black] (xj1);\n    \\path[-] (yi) edge[draw=black] (xj2);\n    \\path[-] (yi) edge[draw=black] (xj3);\n    \\path[-] (yi) edge[draw=black] (xj4);\n    \n\n\n    \n     \\end{tikzpicture}\n    \n    \\caption{Absorbing vertex $v_t$ in the case $r=2s=4$. The blue connection between classes indicates that those edges belong to the fixed matching in the cluster graph while gray connections indicate using additional edges.}\n    \\label{fig:Step52}\n\\end{figure}\n\nWe continue in a similar fashion to cover $v_2,\\dots,v_s$.\nFor this let $t=2,\\dots,s$ and assume that $i_{t-1}$ is such that $|Y_{i_{t-1}}|-t=|X_{i_{t-1}}| \\equiv 0 \\pmod s$.\nLet $\\mathcal{A}$ be the neighbours of $X_{i_{t-1}}$ in $R$ and let $\\mathcal{B}$ be those clusters which share an edge of $M$ with a cluster from $\\mathcal{A}$.\nBy Fact~\\ref{fact:non-extremal} applied to $\\mathcal{B}$ and $H(v_t)$ there are indices $i_t,j$ such that $X_{i_t} \\in H(v_t)$, $Y_{i_t},X_j,Y_j \\in H$ and $Y_{i_t}Y_j$ and $X_jX_{i_{t-1}}$ are edges of $R$.\nThis gives the path $Y_{i_{t-1}},X_J,Y_j,X_{i_t},Y_{i_t}$ in $R$.\nWe extend the paths $P_{i_t}$ and $P_j$, such that after removing the internal vertices we have $|Y_{i_{t-1}}|=|X_{i_{t-1}}| \\equiv 0 \\pmod s$, $|Y_j|=|X_j|\\equiv 0 \\pmod s$, and $|Y_{i_{t}}|-(t+1)=|X_{i_{t}}|\\equiv 0 \\pmod s$.\nThis can be done similarly as for $P_{i_1}$ above and we add the internal vertices to $W_0$ (this adds $6s$ vertices, see Figure~\\ref{fig:Step52}).\n\nWe repeat the same procedure to cover $v_{2s},\\dots,v_{s+1}$ and are left do deal with $|Y_{i_{s}}|-s=|X_{i_{s}}|\\equiv 0 \\pmod s$ and $|Y_{i_{s+1}}|-s=|X_{i_{s+1}}|\\equiv 0 \\pmod s$.\nFor this let $\\mathcal{A}$ be the clusters in $R$ that share an edge in $M$ with a neighbour of the cluster $Y_{i_j}$ and, similarly, let $\\mathcal{B}$ be the clusters in $R$ that share an edge in $M$ with a neighbour of the cluster $Y_{i_{j+1}}$.\nBy Fact~\\ref{fact:non-extremal} applied to $\\mathcal{A}$ and $\\mathcal{B}$ we find indices $j_1,j_2$ such that $X_{j_1},Y_{j_1},X_{j_2},Y_{j_2} \\in H$ and $Y_{i_{s}}X_{j_1}$, $Y_{j_1}Y_{j_2}$, and $X_{j_2}Y_{i_{s+1}}$ are edges of $R$.\nWe extend the paths $P_{j_1}$ and $P_{j_2}$ such that after removing the internal vertices we have $|X_{j_1}|=|Y_{j_1}|\\equiv 0 \\pmod s$, $|X_{j_2}|=|Y_{j_2}|\\equiv 0 \\pmod s$, $|Y_{i_{s}}|=|X_{i_{s}}|\\equiv 0 \\pmod s$, and $|Y_{i_{s+1}}|=|X_{i_{s+1}}|\\equiv 0 \\pmod s$.\nWe add the internal vertices to $W_0$ (this adds $10s$ vertices, see Figure~\\ref{fig:Step53}).\n\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8, every node\/.style={inner sep=0.2,outer sep=0.2}]\n    \n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yi1) at (-11.2, 1.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yi2) at (-11.2, 2.75) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj11) at (-7.9, 1.5) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj12) at (-7.9, 2) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj13) at (-7.9, 3) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj14) at (-7.9, 3.5) {};\n    \n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj11) at (-5.3, 1.5) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj12) at (-5.3, 2) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj13) at (-5.3, 3) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj14) at (-5.3, 3.5) {};\n    \\node[] at (-4.7, 3.25) {$P_{i_{j_1}}$};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj15) at (-4.5, 1.5) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj16) at (-4.5, 2) {};\n    \n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj21) at (-1.4, 1.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj22) at (-1.4, 1.75) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj23) at (-2, 2.25) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj24) at (-2, 2.75) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj25) at (-1.4, 3.25) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj26) at (-1.4, 3.75) {};\n    \\node[] at (-1.9, 3.5) {$P_{i_{j_2}}$};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj21) at (1.2, 1.5) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj22) at (1.2, 2) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj23) at (1.2, 3) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj24) at (1.2, 3.5) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (ys1) at (4.7, 1.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (ys2) at (4.7, 2.75) {};\n  \n    \n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{i_{s}}$}] (Yi) at (-11.2,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{j_1}$}] (Xj1) at (-7.9,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{j_1}$}] (Yj1) at (-5,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{j_2}$}] (Yj2) at (-1.7,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{j_2}$}] (Xj2) at (1.2,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{i_s+1}$}] (Ys) at (4.7,2.375) {};\n    \n    \\path[-] (Yi) edge[draw=black!20, line width=20pt] (Xj1);\n    \\path[-] (Xj1) edge[draw=blue!20, line width=20pt] (Yj1);\n    \\path[-] (Yj1) edge[draw=black!20, line width=20pt] (Yj2);\n    \\path[-] (Yj2) edge[draw=blue!20, line width=20pt] (Xj2);\n    \\path[-] (Xj2) edge[draw=black!20, line width=20pt] (Ys);\n    \n    \\path[-] (yi1) edge[draw=black] (xj11);\n    \\path[-] (yi1) edge[draw=black] (xj12);\n    \\path[-] (yi1) edge[draw=black] (xj13);\n    \\path[-] (yi1) edge[draw=black] (xj14);\n    \n    \\path[-] (yi2) edge[draw=black] (xj11);\n    \\path[-] (yi2) edge[draw=black] (xj12);\n    \\path[-] (yi2) edge[draw=black] (xj13);\n    \\path[-] (yi2) edge[draw=black] (xj14);\n    \n    \\path[-] (xj11) edge[draw=black] (yj11);\n    \\path[-] (xj12) edge[draw=black] (yj11);\n    \\path[-] (xj11) edge[draw=black] (yj12);\n    \\path[-] (xj12) edge[draw=black] (yj12);\n    \n    \\path[-] (xj13) edge[draw=black] (yj13);\n    \\path[-] (xj14) edge[draw=black] (yj13);\n    \\path[-] (xj13) edge[draw=black] (yj14);\n    \\path[-] (xj14) edge[draw=black] (yj14);\n    \n    \\path[-] (yj15) edge[draw=black] (yj21);\n    \\path[-] (yj15) edge[draw=black] (yj22);\n    \\path[-] (yj15) edge[draw=black] (yj23);\n    \\path[-] (yj15) edge[draw=black] (yj24);\n    \n    \\path[-] (yj16) edge[draw=black] (yj21);\n    \\path[-] (yj16) edge[draw=black] (yj22);\n    \\path[-] (yj16) edge[draw=black] (yj23);\n    \\path[-] (yj16) edge[draw=black] (yj24);\n    \n    \\path[-] (yj21) edge[draw=black] (xj21);\n    \\path[-] (yj21) edge[draw=black] (xj22);\n    \\path[-] (yj22) edge[draw=black] (xj21);\n    \\path[-] (yj22) edge[draw=black] (xj22);\n    \n    \\path[-] (yj25) edge[draw=black] (xj23);\n    \\path[-] (yj25) edge[draw=black] (xj24);\n    \\path[-] (yj26) edge[draw=black] (xj23);\n    \\path[-] (yj26) edge[draw=black] (xj24);\n\n    \\path[-] (xj21) edge[draw=black] (ys1);\n    \\path[-] (xj21) edge[draw=black] (ys2);\n    \\path[-] (xj22) edge[draw=black] (ys1);\n    \\path[-] (xj22) edge[draw=black] (ys2);\n    \\path[-] (xj23) edge[draw=black] (ys1);\n    \\path[-] (xj23) edge[draw=black] (ys2);\n    \\path[-] (xj24) edge[draw=black] (ys1);\n    \\path[-] (xj24) edge[draw=black] (ys2);\n    \n\n     \\end{tikzpicture}\n    \n    \\caption{Balancing $Y_{i_{s}}$ and $Y_{i_{s+1}}$ in the case $r=2s=4$. As before, we indicate the fixed matching in the cluster graph by blue connections and additional edges by gray connections.}\n    \\label{fig:Step53}\n\\end{figure}\n\nIt is easy to ensure that in each of these steps the new end-tuples are always well-connected.\nDuring this procedure when absorbing $2s$ vertices into the paths we added at most $18 s^2$ vertices to $W_0$.\n\n\\smallskip\n\\textbf{Reset after $\\nu n$ iterations.}\nWe can repeat this for $\\nu n$ steps as this still guarantees that $|W_0| \\le 18 s^2 \\nu n$ as required.\nIt follows from the bound on $\\nu$ that no degree in the regular pairs in $M$ dropping below $\\tfrac d2 \\cdot \\tfrac{n}{\\ell}$.\nMoreover, for $i \\in [\\ell']$ the end-tuples of the path $P_i$ that were well-connected at some point still have at least $\\tfrac 14 d^s L$ common neighbours into the respective sets.\nAfter $\\nu n$ steps we want to get close enough to the original situation such that we can continue for another $\\nu n$ steps.\nBy Lemma~\\ref{lem:superreg} for any $i \\in [\\ell']$ we need to remove at most $2 \\eps \\tfrac n\\ell$ vertices from the $X_i,Y_i$ to get that  $(X_i,Y_i)$ is $(2\\eps,\\tfrac{3d}{4})$-super-regular.\nWe will greedily absorb these vertices into the path $P_i$ by alternating between $X_i$ and $Y_i$ without any degree dropping below $\\tfrac{3d}{4} \\cdot \\tfrac n\\ell$.\nHere we use that the original pair was $(\\eps,d)$-regular and that it intersects $W$ in at most $\\tfrac 12 \\tfrac n\\ell$ vertices.\nWhile doing this, we can ensure that, for each $i \\in [\\ell']$, both ends of the path $P_i$ are extended at least two steps and the ends are well-connected again.\nWe add the vertices used in these paths to $W$, also move the vertices from $W_0$ to $W$, and set $W_0=\\emptyset$.\nWe continue by covering $2s$ vertices from $V_0$ as explained above.\n\n\\smallskip\n\\textbf{Covering the last vertices.}\nWe can repeat this until $|V_0|<2s$ and if $|V_0|=0$ we are done with this step.\nOtherwise, we have $|V_0|=t\\not=0$, $n \\not\\equiv 0 \\pmod{2s}$, and we can not find the $\\tfrac r2$-blow-up of a cycle.\nIf $r$ is odd $nr \\equiv 0 \\pmod 2$ implies that $n$ and also $t$ are even.\nWe need to absorb the last $t$ vertices in a different way.\nIf $r$ is even, let $v \\in V_0$ and with Fact~\\ref{fact:non-extremal} pick $j_1,j_4$ such that $X_{j_1},X_{j_4} \\in H(v)$, $Y_{j_1},Y_{j_4} \\in H$, and $X_{j_1}X_{j_4}$ is an edge of $R$.\nThen we consider those clusters that share an edge of $M$ with a neighbour of $Y_{j_1}$ and $Y_{j_4}$ respectively and with Fact~\\ref{fact:non-extremal} pick $j_2,j_3$ such that $X_{j_2},Y_{j_2},X_{j_3},Y_{j_3} \\in H$, and $Y_{j_1}Y_{j_2}$, $Y_{j_4}Y_{j_3}$, and $X_{j_2}X_{j_3}$ are edges of $R$.\nWe extend the path $P_{j_1}$ by following $Y_{j_1},X_{j_1},X_{j_4},Y_{j_4},Y_{j_3},X_{j_3},X_{j_2},Y_{j_2},Y_{j_1}$ such that the vertex $v$ is in the neighbourhood of the new vertices from $X_{j_1}$ and $X_{j_4}$.\nThis allows us to include $v$ into the path.\nNote that this is no longer the $\\tfrac r2$-blow-up of a path (see Figure~\\ref{fig:Step54}).\nIf $r$ is odd let $u,v \\in V_0$ and we proceed similarly to include both vertices (see Figure~\\ref{fig:Step55}).\nHere we find $X_{j_1},X_{j_4} \\in H(v)$ and $X_{i_2},X_{j_3} \\in H(u)$ such that $X_{j_1}X_{j_4}$ and $X_{i_2}X_{j_3}$ are edges of $R$ and then connect $Y_{j_1}$ to $Y_{j_2}$ and $Y_{j_4}$ to $Y_{j_3}$ as in the even case by using four additional clusters for each connection.\nNote that here the path structure on the lower half also is not an $\\tfrac r2$-blow-up (alternating $K_{s,s}$ and $K_{s,s}-K_{1,1}^{(s)}$), but has one edge shifted (alternating $K_{s,s}-K_{1,1}$ and $K_{s,s}-K_{1,1}^{(s-1)}$).\n\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8, every node\/.style={inner sep=0.2,outer sep=0.2}]\n    \n    \n\n    \n \n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj11) at (-5.25, 1) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj12) at (-5.25, 1.5) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj13) at (-4.75, 3) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj14) at (-4.75, 3.5) {};\n    \\node[] at (-5.25, 3.25) {$P_{i_{j_1}}$};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj21) at (-1.7, 2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj22) at (-1.7, 2.75) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj21) at (1.2, 2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj22) at (1.2, 2.75) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj31) at (1.2, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj32) at (1.2, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj31) at (-1.7, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj32) at (-1.7, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj41) at (-5, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj42) at (-5, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj41) at (-7.9, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj42) at (-7.9, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj11) at (-7.9, 2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj12) at (-7.9, 2.75) {};\n\n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{j_1}$}] (Xj1) at (-7.9,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{j_1}$}] (Yj1) at (-5,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{j_2}$}] (Yj2) at (-1.7,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{j_2}$}] (Xj2) at (1.2,2.375) {};\n    \n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$X_{j_4}$}] (Xj4) at (-7.9,-4) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$Y_{j_4}$}] (Yj4) at (-5,-4) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$Y_{j_3}$}] (Yj3) at (-1.7,-4) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$X_{j_3}$}] (Xj3) at (1.2,-4) {};\n    \n    \n    \n\n    \\path[-] (Xj1) edge[draw=blue!20, line width=20pt] (Yj1);\n    \\path[-] (Yj1) edge[draw=black!20, line width=20pt] (Yj2);\n    \\path[-] (Yj2) edge[draw=blue!20, line width=20pt] (Xj2);\n    \n    \\path[-] (Xj4) edge[draw=blue!20, line width=20pt] (Yj4);\n    \\path[-] (Yj4) edge[draw=black!20, line width=20pt] (Yj3);\n    \\path[-] (Yj3) edge[draw=blue!20, line width=20pt] (Xj3);\n    \n    \\path[-] (Xj1) edge[draw=black!20, line width=20pt] (Xj4);\n    \\path[-] (Xj2) edge[draw=black!20, line width=20pt] (Xj3);\n    \n    \\node[circle, draw=red,fill=red, minimum width = 6,  font=\\tiny] (v) at (-10, -0.875) {};\n    \\node[] at (-10.5, -0.875) {$v$};\n    \n    \\path[-] (v) edge[draw=red] (xj11);\n    \\path[-] (v) edge[draw=red] (xj12);\n    \\path[-] (v) edge[draw=red] (xj41);\n    \\path[-] (v) edge[draw=red] (xj42);\n    \n    \\path[-] (xj11) edge[draw=black] (xj41);\n    \\path[-] (xj12) edge[draw=black, bend left = 25] (xj42);\n    \n    \\path[-] (xj21) edge[draw=black, bend right = 20] (xj31);\n    \\path[-] (xj21) edge[draw=black, bend right = 25] (xj32);\n    \\path[-] (xj22) edge[draw=black, bend left = 20] (xj31);\n    \\path[-] (xj22) edge[draw=black, bend left = 25] (xj32);\n    \n    \\path[-] (xj11) edge[draw=black] (yj11);\n    \\path[-] (xj12) edge[draw=black] (yj11);\n    \\path[-] (xj11) edge[draw=black] (yj12);\n    \\path[-] (xj12) edge[draw=black] (yj12);\n    \n    \\path[-] (xj41) edge[draw=black] (yj41);\n    \\path[-] (xj42) edge[draw=black] (yj41);\n    \\path[-] (xj41) edge[draw=black] (yj42);\n    \\path[-] (xj42) edge[draw=black] (yj42);\n    \n    \\path[-] (yj31) edge[draw=black] (yj41);\n    \\path[-] (yj32) edge[draw=black] (yj41);\n    \\path[-] (yj31) edge[draw=black] (yj42);\n    \\path[-] (yj32) edge[draw=black] (yj42);\n    \n    \\path[-] (yj31) edge[draw=black] (xj31);\n    \\path[-] (yj32) edge[draw=black] (xj31);\n    \\path[-] (yj31) edge[draw=black] (xj32);\n    \\path[-] (yj32) edge[draw=black] (xj32);\n    \n    \\path[-] (yj21) edge[draw=black] (xj21);\n    \\path[-] (yj22) edge[draw=black] (xj21);\n    \\path[-] (yj21) edge[draw=black] (xj22);\n    \\path[-] (yj22) edge[draw=black] (xj22);\n    \n    \\path[-] (yj21) edge[draw=black] (yj13);\n    \\path[-] (yj21) edge[draw=black] (yj14);\n    \\path[-] (yj22) edge[draw=black] (yj13);\n    \\path[-] (yj22) edge[draw=black] (yj14);\n    \n\n     \\end{tikzpicture}\n    \n    \\caption{Absorbing $v$ in the even case, where $r=2s=4$. The blue and gray connections represent the matching edges and non-matching edges in the cluster graph again.}\n    \\label{fig:Step54}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n    \\begin{tikzpicture}[scale=0.8, every node\/.style={inner sep=0.2,outer sep=0.2}]\n    \n    \n\n    \n \n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj11) at (-4.75, 1) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj12) at (-4.75, 1.5) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj13) at (-5.25, 3) {};\n    \\node[circle, draw=orange,fill=orange, minimum width = 6,  font=\\tiny] (yj14) at (-5.25, 3.5) {};\n    \\node[] at (-4.75, 3.25) {$P_{i_{j_1}}$};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj21) at (1.5, 2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj22) at (1.5, 2.75) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj21) at (4, 2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj22) at (4, 2.75) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj31) at (4, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj32) at (4, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj31) at (1.5, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj32) at (1.5, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj41) at (-5, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (yj42) at (-5, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj41) at (-7.9, -3.75) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj42) at (-7.9, -4.25) {};\n    \n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj11) at (-7.9, 2.25) {};\n    \\node[circle, draw=black,fill=black, minimum width = 6,  font=\\tiny] (xj12) at (-7.9, 2.75) {};\n\n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{j_1}$}] (Xj1) at (-7.9,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{j_1}$}] (Yj1) at (-5,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$Y_{j_2}$}] (Yj2) at (1.5,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = above:{$X_{j_2}$}] (Xj2) at (4,2.375) {};\n    \n    \n    \n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$X_{j_4}$}] (Xj4) at (-7.9,-4) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$Y_{j_4}$}] (Yj4) at (-5,-4) {};\n    \n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (S1) at (-3.25,-4) {};\n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (S2) at (-2.25,-4) {};\n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (S3) at (-1.25,-4) {};\n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (S4) at (-0.25,-4) {};\n    \n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (T1) at (-3.25,2.375) {};\n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (T2) at (-2.25,2.375) {};\n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (T3) at (-1.25,2.375) {};\n    \\node[ellipse, draw = black, minimum width = 20, minimum height = 50] (T4) at (-0.25,2.375) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$Y_{j_3}$}] (Yj3) at (1.5,-4) {};\n    \n    \\node[ellipse, draw = black, minimum width = 40, minimum height = 100, label = below:{$X_{j_3}$}] (Xj3) at (4,-4) {};\n    \n    \n    \n\n    \\path[-] (Xj1) edge[draw=blue!20, line width=20pt] (Yj1);\n    \n    \n    \\path[-] (Yj1) edge[draw=black!20, line width=10pt] (T1);\n    \\path[-] (T1) edge[draw=blue!20, line width=10pt] (T2);\n    \\path[-] (T2) edge[draw=black!20, line width=10pt] (T3);\n    \\path[-] (T3) edge[draw=blue!20, line width=10pt] (T4);\n    \\path[-] (Yj2) edge[draw=black!20, line width=10pt] (T4);\n    \n    \n    \\path[-] (Yj2) edge[draw=blue!20, line width=20pt] (Xj2);\n    \n    \\path[-] (Xj4) edge[draw=blue!20, line width=20pt] (Yj4);\n    \n    \\path[-] (Yj4) edge[draw=black!20, line width=10pt] (S1);\n    \\path[-] (S1) edge[draw=blue!20, line width=10pt] (S2);\n    \\path[-] (S2) edge[draw=black!20, line width=10pt] (S3);\n    \\path[-] (S3) edge[draw=blue!20, line width=10pt] (S4);\n    \\path[-] (Yj3) edge[draw=black!20, line width=10pt] (S4);\n    \n    \n    \\path[-] (Yj3) edge[draw=blue!20, line width=20pt] (Xj3);\n    \n    \\path[-] (Xj1) edge[draw=black!20, line width=20pt] (Xj4);\n    \\path[-] (Xj2) edge[draw=black!20, line width=20pt] (Xj3);\n    \n    \\node[circle, draw=red,fill=red, minimum width = 6,  font=\\tiny] (v) at (-9.5, -0.875) {};\n    \\node[] at (-10, -0.875) {$v$};\n    \\node[circle, draw=red,fill=red, minimum width = 6,  font=\\tiny] (u) at (5.8, -0.875) {};\n    \\node[] at (6.3, -0.875) {$u$};\n    \n    \\path[-] (v) edge[draw=red] (xj11);\n    \\path[-] (v) edge[draw=red] (xj12);\n    \\path[-] (v) edge[draw=red] (xj41);\n\n   \n    \\path[-] (u) edge[draw=red] (xj32);\n    \\path[-] (u) edge[draw=red] (xj21);\n    \\path[-] (u) edge[draw=red] (xj22);\n    \n    \\path[-] (xj11) edge[draw=black] (xj41);\n    \\path[-] (xj12) edge[draw=black, bend left = 25] (xj42);\n    \n    \\path[-] (xj21) edge[draw=black] (xj31);\n    \\path[-] (xj22) edge[draw=black, bend right = 25] (xj32);\n    \n    \\path[-] (xj12) edge[draw=black] (yj14);\n    \\path[-] (xj11) edge[draw=black] (yj13);\n\n    \\path[-] (xj41) edge[draw=black] (yj41);\n    \\path[-] (xj42) edge[draw=black] (yj41);\n    \\path[-] (xj42) edge[draw=black] (yj42);\n    \n    \\path[-] (yj31) edge[draw=black, dotted] (yj41);\n    \\path[-] (yj31) edge[draw=black, dotted] (yj42);\n    \\path[-] (yj32) edge[draw=black, dotted] (yj42);\n    \n    \\path[-] (yj31) edge[draw=black] (xj31);\n    \\path[-] (yj32) edge[draw=black] (xj31);\n    \\path[-] (yj32) edge[draw=black] (xj32);\n    \n    \\path[-] (yj21) edge[draw=black] (xj21);\n    \\path[-] (yj22) edge[draw=black] (xj22);\n    \n    \\path[-] (yj21) edge[draw=black, dotted] (yj11);\n    \\path[-] (yj21) edge[draw=black, dotted] (yj12);\n    \\path[-] (yj22) edge[draw=black, dotted] (yj11);\n    \\path[-] (yj22) edge[draw=black, dotted] (yj12);\n    \n\n     \\end{tikzpicture}\n    \n    \\caption{Absorbing $v$ and $u$ in the odd case, where $r=3, s=2$. The blue and gray connections represent the matching edges and non-matching edges in the cluster graph again. The dotted edges indicate an $r-$regular $r-$connected path.}\n    \\label{fig:Step55}\n\\end{figure}\n\n\\smallskip\n\\textbf{Summary.}\nWe need to estimate the number of vertices added to $W$ throughout the whole procedure of covering $V_0$, which are at most $2 \\beta n$ vertices.\nThere are at most $\\lceil \\tfrac{2\\beta n}{2s\\nu n} \\rceil \\le \\tfrac{\\beta}{s \\nu} + 1$ iterations of the argument for covering $2s\\nu n$ vertices and for covering $2s$ vertices of $V_0$ we need at most $18 s^2$ vertices.\nDuring these iterations we will always have \n\\[|W| \\le \\frac{18s^2}{2s} 2 \\beta n + \\left(\\frac{\\beta}{s\\nu}+1\\right) \\cdot \\ell' 2 \\eps \\frac n\\ell \\cdot 2s \\le 20 s \\beta n\\,,\\]\nwhere the second term comes from the vertices we need to absorb after each iteration.\nCovering the last $t<2s$ vertices does not change anything and, therefore, we can indeed repeat this until all vertices from $V_0$ are covered by the paths.\n\n\\smallskip\n\\noindent \\textbf{\\ref{step:spanning}.}\nWe fully absorbed $V_0$ into the connecting paths such that the end-tuples are well-connected to the other side of the matching edge.\nLet $i \\in [\\ell']$ and denote by $\\mathbf{x}=(x_1,\\dots, x_s)$ and $\\mathbf{y}=(y_1,\\dots, y_s)$ the end-tuples of the paths $P_{i-1}$ and $P_{i}$, respectively.\nRemove $\\mathbf{x}$ from $X_i$ and $\\mathbf{y}$ from $Y_i$, note that $|X_i|=|Y_i| \\equiv 0 \\pmod s$ and that $(X_i,Y_i)$ is $(2\\eps,\\tfrac d2)$-super-regular.\nDenote the common neighbours of $\\mathbf{x}$ in $Y_i$ by $Y'$, the common neighbours of $\\mathbf{y}$ in $X_i$ by $X'$, and note that $|X'|,|Y'| \\ge \\tfrac 14 d^s |X|$.\nTherefore, we can apply Lemma~\\ref{lem:blowup} to cover $X_i$ and $Y_i$ with the $\\tfrac r2$-blow-up of a path and end-tuples within $X'$ and $Y'$, which then connects $P_{i-1}$ to $P_{i}$.\n\nTogether this gives an $r$-regular subgraph in $G$.\nIn the case when $n \\equiv 0 \\pmod{2s}$ we have constructed the $\\tfrac r2$-blow-up of a path, which is $r$-connected.\nTo see that it is also $r$-connected in the other cases it suffices to observe that in the case when $r$ is even removing a perfect matching from a $K_{s,s}$ and adding a vertex $v$ to all these $r$ vertices (see Figure~\\ref{fig:Step54}) preserves this property.\nSimilarly, in the case when $r$ is odd, removing a perfect matching from two copies of $K_{s,s}$, connecting vertices $u,v$ two $r$ of these vertices, and shifting the path in between as described (see Figure~\\ref{fig:Step55}) also preserves this property.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nAttack trees allow for an effective security analysis by systematically organizing the different \nways in which a system can be attacked into a tree. \nThe root node of an attack tree \nrepresents the {\\em attacker's goal} and the children of a given node represent \nits refinement into {\\em sub-goals}. A refinement is typically either disjunctive \n(denoted by $\\OR$) or conjunctive (denoted by $\\AND$). \nThe \nleaves of an attack tree represent the attacker's actions and are called {\\em basic \nactions}. \n\nSince their inception by Schneier~\\cite{Schn}, attack trees\nhave quickly become a popular modeling tool for security \nanalysts.\nHowever, the limitations of this formalism,\nin particular with respect to expressing the order in which the\nvarious attack steps are executed, have been recognized by\nmany authors (see \\eg,~\\cite{KoPiSc_CSR14}).\nIn practice, modeling of security scenarios often requires constructs\nwhere conditions on the execution order of the attack components can\nbe clearly specified. This is for instance the case when the time or\n(conditional) probability \nof an attack is considered, as\nin~\\cite{Arnold-POST14,LL2011}. Consequently, several studies have\nextended attack trees informally with sequential conjunctive\nrefinements. Such extensions have resulted in improved modeling and\nanalyses (e.g., \\cite{LL2011,PB2010,JW2009}) and\nsoftware tools, e.g., ATSyRA~\\cite{pinchinat:hal-01064645}.\n\nEven though the sequential conjunctive refinement, \nthat we denote by \\SAND, \nis well understood at a conceptual level \nand even applied to real world scenarios~\\cite{pinchinat:hal-01064645}, \nnone of the existing solutions have provided \na rigorous mathematical formalization of attack trees with \\SAND. \nIndeed, the extensions found in the literature \nare rather diverse in terms of application domain, interpretation, and \nformality. Thereby, it is infeasible\n to answer fundamental questions such as: \nWhat is the precise expressibility of \\SAND~attack trees?\nWhen do two such trees \nrepresent the same security \nscenario? Or what type of attributes can be synthesized \non \\SAND~attack trees in the standard bottom-up way?\nThese questions can only be precisely answered if \\SAND~attack trees are \nprovided with a formal, general, and explicit interpretation, that is to say, \nif \\SAND~attack trees are given a formal foundation. \n\n\\noindent{\\em Contributions:} \nIn this article we formalize \nthe meaning of a $\\SAND$ attack tree by defining its semantics. Our \nsemantics \nis based on series-parallel (SP) graphs, which is a well-studied branch of \ngraph theory. We provide a complete \naxiomatization \nfor the SP semantics and show that the \nSP semantics for $\\SAND$ attack trees are a conservative extension of the \nmultiset semantics for standard attack trees~\\cite{MaOo} (i.e., our \nextension \ndoes not \nintroduce unexpected equivalences w.r.t.\\ the multiset semantics). \nTo do so, \nwe define a term \nrewriting system that is terminating and confluent\nand obtain normal forms for $\\SAND$ attack trees.\nAs a consequence, we achieve the rather surprising result that the\ndomains of $\\SAND$ attack trees\nand sets of SP graphs are isomorphic. \nWe also extend the notion of attributes \nfor $\\SAND$ attack trees which enable \nthe quantitative analysis of attack scenarios using the standard bottom-up \nevaluation algorithm. \n\n{\nOne of the goals of our work is to provide a level of abstraction that\nencompasses most of the existing approaches from literature. For example,\noperators, \nsuch as the priority-based and the\ntime-based connectors~\\cite{WaWhPhPa}, are indeed captured by the\n$\\SAND$ operator defined in this article. Moreover, other published\nsemantics, such as those based on cumulative distribution\nfunctions~\\cite{Arnold-POST14}, conditional\nprobabilities~\\cite{WaWhPhPa}, or boolean algebra~\\cite{Khan}, can be\nexpressed as an attribute in our formalism. Last but not least, even\nthough we make the distinction between $\\AND$ and $\\SAND$ refinements\nexplicit, our semantics satisfies backward compatibility with the\nwell-known multiset semantics of attack trees~\\cite{MaOo}.\nThis stresses the, much needed, unifying character of our approach.\n}\n\n\\noindent{\\em Organization:} Section~\\ref{sec:related-work} summarizes the \nrelated work and puts our work in context. \nSection~\\ref{sec:sand-atree} provides a formal definition of \\SAND~attack trees\nand its semantics using series-parallel graphs. Section~\\ref{sec:axioms}\ndefines a complete set of axioms for \\SAND~attack trees and presents \na term rewriting system which allows identification of\nsemantically equivalent \\SAND~attack trees. Section~\\ref{sec:attributes}\noutlines an approach to quantitatively analyze \\SAND~attack trees using \nattributes. Finally, Section~\\ref{sec:conclusions} concludes with an outlook \non future work.\n\n\n\n\n\\section{Related Work and Motivation}\n\\label{sec:related-work}\n\n\n\n{}\nSeveral extensions of attack trees with temporal or \ncausal dependencies between \nattack steps have been proposed.\nWe observe that there are three different approaches to achieve this\ngoal.\nThe first approach is to use standard attack trees with the added\nassumption that the children of an \\AND\\ node are sequentially\nordered. This approach is mostly applied to the design\nof algorithms or tools for the analysis of attack trees under the\nassumption of ordered events. \n\nThe second approach is to introduce a mechanism for ordering\nevents in an attack tree, for instance by adding a\nnew type of edge to express causality or conditionality. In its most\ngeneral case, any partial order on the events in an attack tree can be\nspecified.\nThe third approach consists of the introduction of a new type of\nnode for sequencing. Most extensions fall in this category.\nThis approach is used by authors who require their formalism to be\nbackward compatible, or who need standard, as well as ordered\nconjunction. We discuss for each of these approaches \nthe most relevant papers with respect to the present article.\n{\nThat is, we only consider approaches that still have the main\ncharacteristics of attack trees, being the presence of \\AND\\ and \\OR\\\nnodes and the interpretation of the edges as a refinement relation.\nThus, we consider approaches such as attack\ngraphs~\\cite{SHJLW2002,OBM2006} and Bayesian\nnetworks~\\cite{QL2004,JW2007,LM2005} as out of scope\nfor this paper.\n}\n\n\n\\subsubsection{Approaches with a sequential interpretation of \\AND.}\nIn their work on \\emph{Bayesian networks for security}, Qin and\nLee define a transformation from attack trees to Bayesian\nnetworks~\\cite{QiLe}. They state that ``there always exists\nan implicit dependent and sequential relationship between \\AND\\ nodes\nin an attack tree.'' Most literature on attack trees seem to\ncontradict this statement, implying that there is a need to explicitly\nidentify such sequential relationships.\n\nJ\\\"urgenson and Willemson developed an algorithm to calculate the\n\\emph{expected outcome} of an attack tree~\\cite{JW2009}. The goal of\nthe algorithm is to determine a permutation of leaves for which the\noptimal expected outcome for an attacker can be achieved. In essence,\ntheir input is an attack tree where an \\AND\\ node represents all\npossible sequences of its children.\nA peculiarity of their interpretation is that multiple occurrences of\nthe same node are considered only once, implying that the execution of\ntwice the same action cannot be expressed. \n\n\\subsubsection{Approaches introducing a general order.}\n{\nPeine, Jawurek, and Mandel \nintroduce \\emph{security goal indicator trees}~\\cite{PJM2008}\nin which nodes can be related by a notion of \\emph{conditional\ndependency} and Boolean connectors. \nThe authors, \nhowever, do not formally specify the syntax and \nsemantics of the model. \n} \nA more general approach is proposed by Pi\\`etre-Cambac\\'ed\\`es and \nBouissou \n\\cite{PB2010}, who apply \n\\emph{Boolean logic driven Markov\nprocesses} to security modeling. Their formalism does not introduce new\ngates, but a (trigger-)relation on the nodes of the attack tree.\nAlthough triggers \ncan express a more general sequential relation than the $\\SAND$\noperator, they lack the readability of standard attack tree operators. \n\n\\emph{Vulnerability cause graphs}~\\cite{ArBySh,ByArShDu} combine properties of attack \ntrees ($\\AND$ and $\\OR$ nodes) and attack graphs (edges express order \nrather \nthan refinement). The interaction between the $\\AND$ nodes and the order \nrelation is defined through a graph transformation called \\emph{conversion of \nconjunctions}, which ignores the order between nodes. This \ndiscrepancy could be solved by considering distinct\nconjunctive and sequential conjunctive nodes, as we do in this paper.\n\n\\subsubsection{Approaches introducing sequential \\AND.}\nAs noted by Arnold et \nal.~\\cite{Arnold-POST14}, the analysis of time-dependent \nattacks requires attack trees to be extended with a sequential operator. This \nis\naccomplished by defining sequential nodes as conjunctive nodes with a\nnotion of progress of time.\nThe authors define a formal semantics for this extension based on cumulative\ndistribution functions (CDFs), where a CDF denotes the probability that a\nsuccessful attack occurs within time $t$. The main difference with our work is \nthat their approach is based on\nan explicit notion of time, while we have a more abstract approach\nbased on causality. In their semantics, the meaning of an extended\nattack tree is a CDF, in which the relation to the individual basic\nattacks is not explicit anymore. In contrast, in our semantics\nthe individual basic attacks and their causal ordering remain visible.\nAs such, our semantics can be considered more abstract,\nand indeed, we can formulate their semantics as an \\emph{attribute} in our\napproach.\n\n\n\\emph{Enhanced attack trees}~\\cite{AY2006} (EATs) distinguish between $\\OR$, $\\AND$ \nand $\\OAND$ (Ordered $\\AND$). Similarly to the approach of Arnold et\nal.~\\cite{Arnold-POST14}, ordered $\\AND$ nodes are used to express \ntemporal dependencies between attack components. The authors evaluate EATs by \ntransforming them into tree automata. Intermediate states in the automaton \nsupport the task of reporting partial attacks. However, because every \nintermediate node of the tree corresponds to a state in the tree automaton, \ntheir approach does not scale well. This problem can be addressed by \nconsidering \nthe normal form of attack trees, as proposed in this article. \n\nNot every extension of attack trees with $\\SAND$ refinements concerns \ntime-dependent attack scenarios; some aim at supporting risk \nanalyses with conditional\nprobabilities. For that purpose, Wen-Ping and Wei-Min \nintroduce \\emph{improved attack trees}~\\cite{LL2011}. The concepts, \nhowever, are described at an intuitive level only. \n\n\\emph{Unified parameterizable attack trees}~\\cite{WaWhPhPa} unify different extensions \nof attack trees (structural, computational, and hybrid). The authors \nconsider two types of ordered $\\AND$ connectors: priority-based connectors\nand time-based connectors. \nThe children of the former are ordered from\nhighest to lowest priority, whereas the children of the latter are\nordered temporally. \nOur formalism gives a single interpretation to the \n$\\SAND$ operator, yet it can capture both connectors. \n\n{\nDue to obvious similarities, we also review approaches that \nintroduce the $\\SAND$ operator in fault \ntrees. For example, Brooke and Paige include\nfive fault tree gates: $\\AND$, $\\OR$, priority $\\AND$, exclusive $\\OR$,\nand an inhibit gate~\\cite{BP2003}. The authors do not\ndiscuss the semantics of their model for security, though. Another fault tree \nbased approach is discussed by\nKhand~\\cite{Khan}, who proposes to extend attack trees with a set of gates\nfrom dynamical fault tree modeling that overlaps with the gates used\nby Brooke and Paige~\\cite{BP2003} and in particular contains the priority \n$\\AND$ gate. \nKhand assigns truth values to his \nattack trees by giving truth tables for all gates. Khand's\ntruth tables, when restricted to $\\AND$, $\\OR$, and priority $\\AND$, \nconstitute an attribute domain \nwhich is compatible (in the sense\nof~\\cite{KoMaRaSc_JLC}) with the SP semantics for \n$\\SAND$ attack trees as defined in this paper.\n}\n\nWe observe that the extensions of attack trees with sequential conjunction are \nrather diverse in terms of application domain, interpretation, and formality. \nIn order to give a clear and unambiguous interpretation of the $\\SAND$ operator \nand capture different application domains, it is necessary to give a formal \nsemantics as a translation to a well-understood domain. Note that, neither the \nmultiset~\\cite{MaOo} nor the propositional \nsemantics~\\cite{KoPoSc} can express ordering of attack components. \nTherefore, a richer semantical domain needs to be defined. \nThe purpose of this article is to address this problem. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Attack Trees with Sequential Conjunction}\n\\label{sec:sand-atree}\n\nWe extend the attack tree formalism so that a refinement of a\n(sub-)goal of an attacker can be a sequential conjunct (denoted by\n\\SAND) in addition to disjuncts and conjuncts. We first give a \ndefinition of attack trees with the new sequential operator and then \ndefine series-parallel graphs on which the semantics for the new attack\ntrees is based.\n\n\\subsection{\\SAND \\ Attack Trees}\\label{sec:atree-sand}\n\nLet $\\basicact$ denote the set of all possible basic actions of an attacker. \nWe formalize standard attack trees introduced by Schneier in~\\cite{Schn} \nand call them simply \\emph{attack trees} in the rest of this\npaper. Attack trees are closed terms \nover the signature $\\basicact\\cup \\{\\OR, \\AND\\}$, \ngenerated \nby the following grammar, where $b\\in\\basicact$ is a terminal symbol. \n\\begin{equation} \n\\label{gram:AT}\nt ::= b \\mid \\OR(t, \\dots, t) \\mid \\AND(t, \\dots, t).\n\\end{equation}\nThe universe of attack trees is denoted by $\\atree$. \n\\emph{$\\SAND$ attack trees} are closed terms over the signature\n$\\basicact\\cup \\{\\OR, \\AND, \\SAND\\}$, where \n$\\SAND$ is a non-commutative operator called \\emph{sequential conjunction},\nand are generated by the grammar\n\\begin{equation} \n\\label{gram:SAT}\nt ::= b \\mid \\OR(t, \\dots, t) \\mid \\AND(t, \\dots, t) \\mid \\SAND(t, \\dots, t\n). \n\\end{equation}\nThe universe of $\\SAND$ attack trees is denoted by $\\sandtree$. The \npurpose of \\OR\\ and \\AND\\ refinements in $\\SAND$ attack trees\nis the same as in \nattack trees. The sequential conjunctive refinement \\SAND\\ allows us to model \nthat a certain goal is reached if and only if all \nits subgoals are reached in a precise order. \n\nThe following attack scenario \nmotivates the need for extending attack trees with sequential\nconjunctive refinement. \n\\begin{example}\n\\label{eg:def-attack-tree}\nConsider \na file server $\\server$, offering\nftp, ssh, and rsh services. \nThe attack tree in Figure~\\ref{fig:attack-tree}\nshows how an attacker can gain root privileges on \n$\\server$ (\\becomeroot), in two ways: \neither without providing any user credentials \n(\\noauth)\nor by breaching the authentication mechanism\n(\\auth).\n\\begin{figure}[t]\n\\centering\n\\includegraphics[height=4.5cm]{atree1}\n\\caption{An attack tree with sequential and parallel conjunctions}\n\\label{fig:attack-tree}\n\\end{figure}\n\nIn the first case,\nthe attacker must first gain user privileges on $\\server$ \n(\\textit{gain user pri\\-vi\\-leges}) \nand then perform a local buffer overflow attack (\\localbofx). \nSince the attack steps must be executed in this particular order, \nthe use of \\SAND~refinement is substantial.\nTo gain user privileges, the attacker \nmust exploit an ftp vulnerability \nto anonymously upload a list of trusted hosts to $\\server$ \n(\\ftprhostsx).\\footnote{For readability, attack actions are named after\nthe services that are exploited.} Finally, she can use the\nnew trust condition to remotely execute shell commands on $\\server$ (\\rshx). \n \nThe second way is to abuse a buffer overflow in both the \nssh daemon (\\sshbofx) and the RSAREF2 library (\\rsarefbofx) used for authentication.\nThese attacks can be executed in any order, which is modeled with \nthe standard \\AND\\ refinement. \n\nUsing the term notation introduced in this section, \nwe can represent the \\SAND\\ attack tree in Figure~\\ref{fig:attack-tree} \nas\n\\[\nt = \\OR\\Big(\\SAND\\big(\\SAND(\\ftprhostsx, \\rshx), \\localbofx\\big), \\\\\n \\AND(\\sshbofx, \\rsarefbofx)\\Big),\n\\]\nwhere $\\ftprhostsx, \\rshx, \\localbofx, \\sshbofx, \\rsarefbofx\\in \\basicact$\nare basic actions.\n\\end{example}\n\n\\subsection{Series-Parallel Graphs}\n\\label{sec:spgraphs}\n\nA \\emph{series-parallel graph} (SP graph) is an edge-labeled directed graph that has \ntwo unique, distinct vertices, called \\emph{source} and \\emph{sink}, \nand that can be constructed with\nthe two operators for sequential and parallel composition of graphs\nthat we formally define below. \nA source is a vertex which has no incoming edges and a \nsink is a vertex without outgoing edges. \n\nOur formal definition of SP graphs is based on \\emph{multisets},\ni.e., sets in which members are allowed to occur more than once. \nWe use $\\msl \\cdot \\msr$ to denote multisets and $\\powerset(\\cdot)$\nto denote powersets. \nThe \\emph{support} $M^{\\star}$ of a multiset $M$ is the set of \ndistinct elements in $M$. \nFor instance, the support of the multiset $M = \\msl b_1, b_2, b_2 \\msr$\nis $M^{\\star} = \\{b_1, b_2\\}$. \n\nIn order to define SP graphs, we first introduce the notion of \nsource-sink graphs labeled by the elements of $\\basicact$. \n\n\\begin{definition}\nA \\emph{source-sink graph} over $\\basicact$ is a tuple\n$G=(V,E,s,z)$, where $V$ is the set of vertices,\n$E$ is a multiset of labeled edges with support $E^{\\star} \\subseteq V \\times \n\\basicact \\times V$, \n$s\\in V$ is the unique source, $z\\in V$ is the unique sink, \nand $s\\not =z$. \n\\end{definition}\nThe sequential composition of a source-sink graph $G=(V,E,s,z)$ with a \nsource-sink graph \n$G'=(V',E',s',z')$, denoted by $G\\seqop G'$, is the\ngraph resulting from taking the \ndisjoint union of $G$ and $G'$ and identifying the\nsink of $G$ with the source of $G'$. \nMore precisely,\nlet $\\dot\\cup$ denote the disjoint union operator\nand $E^{[s\/z]}$ denote the multiset of edges in $E$, where all occurrences of \nvertex $z$ are replaced by vertex $s$. \nThen we define\n\\[G\\seqop G' = (V\\setminus\\{z\\}\\dot\\cup V', E^{[s'\/z]}\\dot\\cup E', s,\nz').\\]\nThe parallel composition, denoted by $G\\parop G'$, is defined\nsimilarly, except that the two sources are identified \nand the two sinks are identified. \nFormally, we have\n\\[G\\parop G' = (V\\setminus\\{s,z\\}\\dot\\cup V', E^{[s'\/s,z'\/z]}\\dot\\cup\nE', s', z').\\] \nIt follows directly from the definitions that \nthe sequential composition is associative and that the \nparallel composition is associative and\ncommutative.\n\nWe write $\\xrightarrow{b}$ for the graph with a single edge labeled with \n$b$ and define SP graphs as follows. \n\\begin{definition} \\label{def:sp-graph}\nThe set $\\spset$ of \\emph{series-parallel graphs} (SP graphs) over \n$\\basicact$ is defined inductively by the following two rules\n\\begin{itemize}\n\\item For $b\\in \\basicact$, $\\xrightarrow{b}$ is an SP graph.\n\\item If $G$ and $G'$ are SP graphs, then so are \n$G\\seqop G'$ and $G\\parop G'$.\n\\end{itemize}\n\\end{definition}\nIt follows directly from Definition~\\ref{def:sp-graph} \nthat SP graphs are connected and acyclic. Moreover, \nevery vertex of an SP graph lies on a path from the source to\nthe sink. \nWe consider two SP graphs to be \\emph{equal} \nif there is a bijection between their\nsets of vertices that preserves the edges and edge labels.\n{\n}\n\\begin{example}\nFigure~\\ref{fig:example-spgraph} shows an example of an SP graph with\nthe source $s$ and the sink $z$.\nThis graph corresponds to the construction\n\\[\\big(\\xrightarrow{a}\\parop \\xrightarrow{b} \\parop \\xrightarrow{b}\\big) \\seqop\n\\xrightarrow{c} \\seqop \\Big(\\big(\\xrightarrow{d} \\seqop (\\xrightarrow{e}\\parop\n\\xrightarrow{f})\\big) \\parop \\xrightarrow{g}\\Big).\\]\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[height=2.2cm]{sp-graph}\n\\caption{A series-parallel graph}\n\\label{fig:example-spgraph}\n\\end{figure}\n\n\\end{example}\n\n\\subsection{SP Semantics for $\\SAND$ Attack Trees}\n\\label{sec:atree-sand-semantics}\n\nNumerous semantics have been proposed to \ninterpret attack trees, \nincluding \npropositional logic~\\cite{KoPoSc_iFM'14}, multisets~\\cite{MaOo}, \n De Morgan lattices~\\cite{KoPoSc}, tree automata~\\cite{AY2006},\nand \nMarkov processes~\\cite{PB2010,Arnold-POST14}. \nThe choice of a semantics allows us to accurately represent the\nassumptions made in a security scenario, e.g., \nwhether actions can be repeated or resources reused, and \nto decide which trees represent the same security scenario. \nThe advantages of formalizing attack trees and the need for various\nsemantics have been discussed in~\\cite{KoMaRaSc_JLC}.\nSince attack trees are \\AND\/\\OR~trees, the most natural interpretation \nis based on propositional logic. However, because the logical operators \nare idempotent, this interpretation assumes \nthat the multiplicity of an action is irrelevant. \nAs a consequence, \nthe propositional semantics is not well suited to reason about \nscenarios with multiple occurrences of the same action. \nDue to this lack of expressivity a semantics was\nproposed~\\cite{MaOo} in which \nthe multiplicity of actions is taken into account. This was achieved\nby interpreting an attack tree as a set of multisets that represent\ndifferent ways of reaching the root goal. \nThis multiset semantics is compatible\nwith computations \nthat depend on the number of occurrences of an action in the tree, such as \nthe minimal time to carry out the attack represented by the root goal. \n \nWe now extend the multiset semantics to $\\SAND$ attack trees. \nSince SP graphs naturally extend multisets with \na partial order, they supply a formalism in which we can interpret \ntrees using both --- commutative and sequential --- conjunctive refinements. \nSP graphs therefore provide a canonical semantics for \\SAND\\ trees in which \nmultiplicity and ordering of goals and actions are significant.\nThe idea is to interpret an attack tree $t$ as a set of SP graphs. \nThe semantics $\\sem{t}=\\{G_1,\\dots,G_k\\}$ \nof a tree $t$ corresponds to the set of possible \nattacks $G_i$, where each attack is\ndescribed by an SP graph labeled by the basic actions of $t$. \n{\n}\n\n\\begin{definition} \\label{def:sp-sem}\nThe \\emph{SP semantics} for $\\SAND$ attack trees is given by the function \n$\\sem{\\cdot}: \\sandtree \\to \\power{\\spset}$, \nwhich is defined recursively as follows:\nfor $b\\in\\basicact$, $t_i \\in \\sandtree$, $1\\leq i \\leq k$, \n\\[\n\\def1.3{1.3}\n\\begin{array}{l}\n\\sem{b} = \\{\\xrightarrow{b}\\}\\\\\n\\sem{\\OR(t_1, \\dots, t_k)} = \\bigcup_{i=1}^{k}{\\sem{t_i}}\\\\\n\\sem{\\AND(t_1, \\dots, t_k)} =\n \\{ G_1 \\parop \\dots \\parop G_k \\ \\mid \\ \n (G_1, ..., G_k) \\in \\sem{t_1} \\times ... \\times \\sem{t_k}\n \\}\\\\\n\\sem{\\SAND(t_1, \\dots, t_k)} =\n \\{ G_1 \\seqop \\dots \\seqop G_k\\ \\mid \\ \n (G_1, ..., G_k) \\in \\sem{t_1} \\times ... \\times \\sem{t_k}\n \\}.\\\\\n\\end{array}\n\\]\n\\end{definition} \n{\n The SP semantics maps $\\SAND$ attack trees to sets of SP graphs as follows. \n A leaf corresponding to a basic action $b$\n is translated into a singleton set containing the SP graph \n which consists of a single edge labeled with $b$.\n The semantics of a disjunctive node is the set \n of all the alternative attacks described by the node's children.\n The semantics of a conjunctive node is \n the parallel composition of every attack\n alternative from each of its children. \n Finally, the semantics of a sequential conjunctive node is \n a sequential composition of \nattack alternatives for the children. \n}\n\n\\begin{example}\n\\label{ex:sem(t)}\nThe SP semantics of the attack tree $t$ \ndepicted in Figure~\\ref{fig:attack-tree} is\n\\begin{equation*}\n\\sem{t}=\\{\\xrightarrow{\\ftprhostsx}\\xrightarrow{\\rshx}\n\\xrightarrow{\\localbofx}\\ , \\ \n\\xrightarrow{\\sshbofx} \\parop \\xrightarrow{\\rsarefbofx}\\}.\n\\end{equation*}\n\\end{example}\nAs shown in Example~\\ref{ex:sem(t)}, the SP semantics provides an alternative \ngraph representation for attack trees \nand therefore contributes\na different perspective on an attack scenario. \nThe \\SAND~attack tree emphasizes the refinement of goals, whereas \nSP graphs highlight the sequential aspect of attacks.\n{\n}\n\nThe SP semantics provides a natural partition of\n$\\sandtree$ into equivalence classes. \n\\begin{definition}\\label{def:equivalent}\nTwo \\SAND\\ attack trees $t_1$ and $t_2$ are \\emph{equivalent with respect \nto the SP semantics}\nif and only if they are interpreted by the same set of SP graphs, i.e., \n$\\sem{t_1}=\\sem{t_2}$.\n\\end{definition}\nBy Definition~\\ref{def:equivalent}, if the SP semantics provides \naccurate assumptions for an attack scenario, then \ntwo $\\SAND$ attack trees \nrepresent the same attack scenario \nif and only if they are equivalent \nwith respect to the SP semantics. \n\n{We finish this section by noticing that in the case of \\SAND\\ attack \ntrees without any \n$\\SAND$ refinement, the SP semantics coincides with the multiset semantics \nintroduced in~\\cite{MaOo}. Indeed, it suffices to identify the multiset \n$\\msl b_1, \\dots, b_k \\msr$ with the SP \ngraph $\\xrightarrow{b_1}\\parop \\dots \\parop \\xrightarrow{b_k}$. \nWe discuss this issue more in details in Section~\\ref{sec:SP_vs_M}.\n\n}\n\n\n\n\n\n\\section{Axiomatization of the SP Semantics} \n\\label{sec:axioms}\n{\nIn order to provide efficient analysis methods \nfor attack tree-like models, we need to be able to decide whether two \ntrees are equivalent with respect to a given semantics. \nIdeally, we would like to find the most efficient (e.g., the smallest)\nrepresentation of a given security scenario. However, \nin the case of the \\SAND\\ semantics, \nthere exists an infinite number of trees $t'$ equivalent to a given tree $t$.\n\nIn this section we study the mathematical implications of \nusing sets of SP graphs as an interpretation domain for \\SAND\\ attack trees. \nWe introduce an axiomatization of $\\SAND$ attack\ntrees which is complete with respect to the SP semantics.\nThis allows us to reason directly on $\\SAND$ attack trees, without having to\nmove to the semantical domain. Further, we derive a term\nrewriting system from the axiomatization as a means to effectively\ndecide whether two $\\SAND$ attack trees are equivalent with respect to the \nSP semantics. As a\nconsequence, we obtain a canonical representation of $\\SAND$\nattack trees which we prove to be isomorphic to sets of SP graphs.\n}\n\n\n\n\\subsection{A complete set of axioms for the SP semantics}\n\nLet $\\var$ be a set of variables denoted \nby capital letters. \nFollowing the approach developed in~\\cite{KoMaRaSc_JLC}, we \naxiomatize\n$\\SAND$ attack trees \nwith equations \n$l=r$, where $l$ and $r$ are terms over \nvariables in $\\var$, \nconstants in $\\basicact$, and the operators \\AND, \\OR, and \\SAND. \nThe equations formalize the intended properties of refinements\nand provide semantics-preserving transformations of \n\\SAND\\ attack trees. \n\\begin{example}\n\\label{ex:axiom}\nLet $\\mathrm{Sym}_{\\ell}$ denote the set of all bijections\nfrom $\\{1,\\dots,{\\ell}\\}$ to itself. \nThe axiom \n\\[\\AND(Y_1,\\dots,Y_{\\ell})=\\AND(Y_{\\sigma(1)},\\dots,Y_{\\sigma({\\ell})})\\] \nexpresses that the order between children refining a parallel conjunctive node \nis not relevant. \nIn other words, the operator \\AND\\ is commutative. This \nimplies that any two trees of the form \n$\\AND(t_1,\\dots,t_l)$ and $\\AND(t_{\\sigma(1)},\\dots,t_{\\sigma(l)})$\nrepresent the same scenario.\n\\end{example}\n\nOur goal is to define a complete set of axioms, denoted by $\\ESP$, for\nthe SP semantics for \\SAND\\ attack trees. Intuitively, $\\ESP$\nis a set of equations\nthat can be applied to transform \na \\SAND\\ attack tree \ninto any equivalent \\SAND~attack tree \nwith respect to the SP semantics. \nBefore defining the set $\\ESP$, \nwe formalize the notion of a complete set of axioms for a given \nsemantics for (\\SAND) attack trees, following~\\cite{KoMaRaSc_JLC}. \n\nLet $T(\\var,\\Sigma)$ be the free term algebra\nover the set of variables\n$\\var$ and a signature $\\Sigma$, \nand let $E$ be a set of equations over $T(\\var,\\Sigma)$. \nThe equation $t=t'$, where $t,t'\\in T(\\var,\\Sigma)$,\nis a \\emph{syntactic consequence} of $E$ \n(denoted by $E\\vdash t=t'$) \nif it can be derived from $E$ by application \nof the following rules. \nFor all $t,t',t''\\in T(\\var,\\Sigma)$, $\\rho\\colon \\var\\to T(\\var,\\Sigma)$,\n and $X\\in \\var$: \n\\begin{itemize}\n\\item $E\\vdash t=t$, \n\\item if $t= t' \\in E$, then $E\\vdash t=t'$,\n\\item if $E\\vdash t=t'$, then $E\\vdash t'=t$, \n\\item if $E\\vdash t=t'$ and $E\\vdash t'=t''$, then $E\\vdash t=t''$. \n\\item if \n$E\\vdash t=t'$, then\n$E\\vdash \\rho(t)=\\rho(t')$,\n\\item if $E\\vdash t=t'$, then \n$E \\vdash t'' [t\/X] = t'' [t'\/X]$, \nwhere $t'' [t\/X]$ is the term obtained from $t''$\nby replacing all occurrences of variable $X$ with $t$. \n\\end{itemize}\n\n Let $\\sandtreevar$ denote the set of terms constructed from the set of \nvariables $\\var$, the set of basic actions $\\basicact$ (treated as\nconstants), and operators $\\OR$, $\\AND$ and $\\SAND$.\nLet $\\atreevar$ be the set of terms constructed \nfrom the same parts, except for the operator \\SAND. \nUsing the notion of syntactic consequence, \nwe define a complete set of axioms for a semantics for attack trees. \n\\begin{definition}\n\\label{def\/compl_set_axioms}\nLet $\\semantic{\\cdot}$ be a semantics for attack trees \n(resp. \\SAND\\ attack trees)\nand let $E$ be a set of equations over $\\atree^{\\var}$ \n(resp. $\\sandtreevar$). \nThe set $E$ is a \\emph{complete set of axioms} for $\\semantic{\\cdot}$\nif and only if, for all $t,t'\\in \\atree$ (resp. $\\sandtree$)\n\\[\\semantic{t}=\\semantic{t'} \\iff E\\vdash t=t'. \\]\n\\end{definition}\n\nWe are now ready to give a complete set of axioms for the SP semantics\nfor \\SAND\\ attack trees. These axioms \nallow us to determine whether two visually\ndistinct trees represent the same security \nscenario according to the SP semantics. \n\n\\begin{theorem}\n\\label{th:axioms}\nGiven $k,m\\geq 0$, and $\\ell\\geq 1$, let\n$\\Xvec = X_1, \\ldots, X_k$,\n$\\Yvec = Y_1, \\ldots, Y_{\\ell}$, and\n$\\Zvec = Z_1, \\ldots, Z_m$\nbe sequences of variables.\nLet $\\mathrm{Sym}_{\\ell}$ be the set of all bijections\nfrom $\\{1,\\dots,{\\ell}\\}$ to itself. The following \nset of equations over $\\sandtreevar$, denoted by $\\ESP$, is a complete \nset of axioms\\footnote{Note that the axioms are in fact \\emph{axiom schemes}. \nThe operators $\\OR$, $\\AND$ and $\\SAND$ are \\emph{unranked},\nrepresenting infinitely many $k$-ary function symbols ($k\\geq1$).} for \nthe SP semantics for $\\SAND$ attack trees.\n\\begingroup\n\\allowdisplaybreaks\n\\begin{align}\n& \\OR(Y_1,\\dots,Y_{\\ell})=\\OR(Y_{\\sigma(1)},\\dots,Y_{\\sigma({\\ell})}),\n\\quad \\forall \\sigma\\in \\mathrm{Sym}_{\\ell} \\tag{$E_{1}$} \\label{E1}\\\\\n& \\AND(Y_1,\\dots,Y_{\\ell})=\\AND(Y_{\\sigma(1)},\\dots,Y_{\\sigma({\\ell})}),\n\\quad \\forall \\sigma\\in \\mathrm{Sym}_{\\ell} \n\\tag{$E_{2}$}\\label{E2}\\\\ \n& \\OR\\big(\\Xvec,\\OR(\\Yvec)\\big)=\n\\OR(\\Xvec,\\Yvec) \\tag{$E_{3}$} \\label{E3}\\\\\n& \\AND\\big(\\Xvec,\\AND(\\Yvec)\\big)=\n\\AND(\\Xvec,\\Yvec) \\tag{$E_{4}$} \\label{E4}\\\\\n& \\SAND\\big(\\Xvec,\\SAND(\\Yvec),\\Zvec\\big)=\n\\SAND(\\Xvec,\\Yvec,\\Zvec) \\tag{$E_{4'}$} \\label{E4'}\\\\\n& \\OR(A)= A \\tag{$E_{5}$} \\label{E5}\\\\\n& \\AND(A)= A \\tag{$E_{6}$} \\label{E6}\\\\\n& \\SAND(A)= A \\tag{$E_{6'}$} \\label{E6'}\\\\\n& \\AND\\big(\\Xvec,\\OR(\\Yvec)\\big)= \n\\OR\\big(\\AND(\\Xvec,Y_1),\\dots,\\AND(\\Xvec,Y_{\\ell})\\big) \\tag{$E_{10}$} \\label{E10}\\\\\n& \\SAND\\big(\\Xvec,\\OR(\\Yvec),\\Zvec\\big)= \n\\OR\\big(\\SAND(\\Xvec,Y_1,\\Zvec),\\dots,\\SAND(\\Xvec,Y_{\\ell},\\Zvec)\\big) \\tag{$E_{10'}$} \n\\label{E10'}\\\\\n& \\OR(A,A,\\Xvec)= \\OR(A,\\Xvec). \\tag{$E_{11}$} \n\\label{E11}\n\\end{align}\n\\endgroup\n\\end{theorem}\nThe numbering of the axioms in $\\ESP$ corresponds to the numbering \nof the axioms for the multiset semantics for standard attack trees, \nas presented in~\\cite{KoMaRaSc_JLC}, while new axioms (involving \\SAND) are \nmarked with primes. \n\\begin{proof}\nThe proof of this theorem follows the same\nline of reasoning as the proofs of Theorems~4.2 and~4.3 of\nGischer~\\cite{Gischer1988199}, where series--parallel pomsets are axiomatized.\n{\n To prove the theorem, we remark that \n SP graphs form a visual representation of \n series-parallel \n partially ordered multisets (SP pomsets). \n A complete, finite axiomatization of pomsets under \n concatenation, parallel composition and union\n has been provided in~\\cite{Gischer1988199}, where \n sets of series-parallel pomsets have been used to represent processes. \n In our case, sets of series-parallel pomsets (\\ie, sets of SP graphs)\n represent attack trees constructed using \n \\AND\\ (having the same properties as the parallel composition of processes), \n \\SAND\\ (having the same properties as concatenation),\n and \\OR\\ (having the same properties as choice). \n The set $\\ESP$ corresponds to the axioms from~\\cite{Gischer1988199}. \n The axioms involving the identity elements (\\ie, $1$ -- the empty pomset, and \n$0$ --\n the empty process) have been omitted because they can only be used \n for transforming processes involving $0$ or $1$ and \n such identity elements do not exist in the case of attack trees. \n Furthermore, our axioms are written using unranked operators \n contrary to the binary operators of concatenation, parallel composition, and \n choice. \n}\n\n\\qed\n\\end{proof}\n\n\n\\subsection{\\SAND~Attack Trees in Canonical Form}\n\\label{sec:NF}\nLet $\\semantic{\\cdot}$ be a semantics for (\\SAND) attack trees.\nA complete axiomatization of $\\semantic{\\cdot}$\ncan be used to derive a canonical form of trees \ninterpreted with $\\semantic{\\cdot}$. \nSuch canonical forms provide the most concise \nrepresentation for equivalent trees\nand are the natural representatives \nof equivalence classes defined by \n$\\semantic{\\cdot}$. \n\nWhen \\SAND~attack trees are interpreted using the SP semantics, \ntheir canonical forms \nconsist of either a single basic action, \nor of a root node labeled with \\OR\\ and subtrees \nwith nested, alternating \noccurrences of \\AND\\ and \\SAND\\ nodes. \nCanonical forms correspond exactly to \nthe sets of SP graphs labeled by $\\basicact$\nand they depict all attack alternatives in a straightforward \nway.\n\n{\nCanonical representations of \\SAND~attack trees\nunder the SP semantics\ncan be defined using the complete set of axioms \n$\\ESP$. \nBy orienting the equations \n\\eqref{E3}, \\eqref{E4}, \\eqref{E4'}, \\eqref{E5}, \\eqref{E6},\n\\eqref{E6'}, \\eqref{E10}, \\eqref{E10'}, and \\eqref{E11}\nfrom left to right, we obtain a term rewriting system, \ndenoted by $\\thetrs$. \nThe canonical representations of \\SAND~attack trees \ncorrespond to normal forms with respect to $\\thetrs$. \nIn the rest of this section we show that \nthe normal forms with respect to $\\thetrs$\nare exactly the terms generated by the following grammar, \nwhere $k\\geq 2$ and $b\\in\\basicact$\n\\begin{align} \n\\label{grammar:NF}\n\\nfterm ::= \\quad& \\asterm \\mid \\OR(\\asterm_1, \\dots, \\asterm_k)\n \\quad\\mbox{for } \\asterm_i \\neq \\asterm_j\n \\mbox{ if } i\\neq j\\\\\\notag\n\\asterm ::= \\quad & \\andterm \\mid \\sandterm\\\\\\notag\n\\andterm ::= \\quad & b \\mid \\AND(\\sandterm_1, \\ldots, \\sandterm_k)\\\\\\notag\n\\sandterm ::= \\quad& b \\mid \\SAND(\\andterm_1, \\ldots, \\andterm_k).\\\\\\notag\n\\end{align}\nThe non-terminal $\\andterm$ produces all trees that consist of\na single basic action or being \na nested alternation of \n$\\AND$ and $\\SAND$ operators,\nwhere the outer operator is $\\AND$.\nSimilarly, $\\sandterm$ produces all such trees \nwhere the outer operator is $\\SAND$.\nThe non-terminal $\\asterm$ generates the two \npreviously described types of trees. \nFinally, $\\nfterm$ \ncombines the trees generated by $\\asterm$ using \nthe \\OR\\ refinement. \nWe denote the sets of terms \ngenerated by $\\nfterm$, $\\asterm$, $\\andterm$, and $\\sandterm$, by $\\nfterms$, \n$\\asterms$, $\\andterms$, and $\\sandterms$, respectively. \n\nWe first observe that the terms generated by the non-terminal $\\nfterm$\ncorrespond exactly to all sets of SP graphs labeled by the elements of \n$\\basicact$. \n\\begin{lemma}\\label{lem:bijection}\nThe restriction of function $\\sem{.}$ to $\\nfterms$\nis a bijection from $\\nfterms$ to $\\power{\\spset}$.\n\\end{lemma}\n\\begin{proof}\nThe proof consists of two steps. First we prove that the terms from\n$\\asterms$ exactly correspond to SP graphs, after which we extend this\nresult to the correspondence between $\\nfterms$ and the sets of SP graphs.\n\\begin{enumerate}\n\\item $\\sem{.}$ is a bijection from $\\asterms$ to $\\spset$.\\\\\n For injectivity we prove by induction  that $\\sem{\\asterm_1} = \\sem{\\asterm_2}$\n implies $\\asterm_1 = \\asterm_2$. Given that for every $t \\in \\asterms\n $ the set $\\sem{t}$ contains a single SP graph, we abuse notation and \n refer to \n $\\sem{t}$ as the SP graph contained in $\\sem{t}$. Let us assume that \n $\\asterm_1$ is a basic action $b$, then $\\sem{\\asterm_1}$ is a single \n edge graph $\\xrightarrow{b}$, it follows that $\\asterm_2 = \n b$ considering the edge labels preservation property of two isomorphic SP \n graphs. Let us consider now that $\\asterm_1 = \\AND(\\sandterm_1^1, \\cdots, \\sandterm_k^1)$ for \n some $k \\geq 2$, which means that $\\sem{\\asterm_1} = \n \\sem{\\sandterm_1^1} \\parop \\cdots \\parop\n \\sem{\\sandterm_k^1}$. Given that $\\sem{\\asterm_1} = \\sem{\\asterm_2}$,\n for every $i \\in \\{1, \\cdots, k\\}$, $\\sem{\\asterm_2}$ contains a subgraph \n $G_i$ which is isomorphic to $\\sem{\\sandterm_i^1}$. Thus, considering that \n $\\sem{\\sandterm_i^1}$ is either a basic action or a sequential composition, \n we have that \n $\\sem{\\asterm_2} = G_1 \\parop \\cdots \\parop G_k$ and, according to grammar\n \\eqref{grammar:NF}, \n $\\asterm_2 = \\AND(\\sandterm_1^2, \\cdots, \\sandterm_k^2)$, for some terms \n $\\sandterm_1^2, \\cdots, \\sandterm_k^2 \\in \\sandterms$. By the \n induction hypothesis, it \n follows that $\\sem{\\sandterm_i^1} = G_i = \n \\sem{\\sandterm_i^2}$ implies $\\sandterm_i^1 = \\sandterm_i^2$, which leads to \n $\\asterm_1 = \\asterm_2$. A similar \n proof can be obtained for the case where $\\asterm_1 = \\SAND(\\andterm_1^1, \n \\cdots, \\andterm_k^1)$. \n \n\n Surjectivity follows from the fact that every SP graph has a unique\n (modulo associativity) decomposition in terms of \n the operators for \n sequential and parallel composition.\n Such a decomposition naturally corresponds to terms from $\\asterms$.\n\\item $\\sem{.}$ is a bijection from $\\nfterms$ to $\\powerset(\\spset)$.\\\\\n For injectivity, we assume that $\\sem{\\nfterm_1} = \\sem{\\nfterm_2}$.\n This implies that the sets $\\sem{\\nfterm_1}$ and $\\sem{\\nfterm_2}$\n have the same size and the same elements. If this size is $1$, then\n they both contain the same element, which uniquely corresponds to\n a term $\\asterm$, so $\\nfterm_1 = \\asterm = \\nfterm_2$.\n If the size of the sets is larger than $1$, then $\\nfterm_i$ \n are of the form $\\OR(\\asterm_1^i,\\dots, \\asterm_k^i)$, for \n $i\\in\\{1,2\\}$. Since $\\sem{\\nfterm_1} = \\sem{\\nfterm_2}$, \n the elements of these two\n sets are pairwise identical.\n Moreover, by definition, \n all arguments in $\\nfterm_i$ are different, which implies \n $k_1=k_2$.\n From the previous item, it follows\n that the elements of $\\sem{\\nfterm_i}$ correspond uniquely \n to $\\asterm_1^i,\\dots, \\asterm_k^i$. \n From the pairwise equality between the\n arguments of the two terms, it follows that $\\nfterm_1$ and\n $\\nfterm_2$ are identical.\n \n For surjectivity, \n let $\\{G_1, \\dots, G_k\\}$ \n be a set of SP graphs. \n It follows from the previous item\n that there exist trees $\\asterm_1, \\dots, \\asterm_k$, \n such that $\\sem{\\asterm_i}=G_i$, for $i\\in\\{1,\\dots, k\\}$. \n This implies that $\\{G_1, \\dots, G_k\\}= \\sem{\\OR(\\asterm_1, \\dots, \\asterm_k)}$, \n which finishes the proof of surjectivity. \n\\end{enumerate}\n\\qed\n\\end{proof}\nLemma~\\ref{lem:bijection} shows that \nthe grammar \\eqref{grammar:NF} \ngenerates all \\SAND~attack trees in canonical form. \nIt remains to be proven that\nthe set of trees generated by the grammar \\eqref{grammar:NF}\nis equal to the set of normal forms of the term rewriting system\n$\\thetrs$ and that these normal forms are unique. \n\n\\begin{theorem}\n\\label{th:trs}\nThe term rewriting system $\\thetrs$ is strongly terminating and confluent.\n\\end{theorem}\n\\begin{proof}\nWe show that the term rewriting system $\\thetrs$ is terminating and confluent\nwith help of the grammar~\\eqref{grammar:NF}, \nin four steps. \n\\begin{enumerate}\n\\item \nFirst, we show with\nstandard methodology that the\nterm rewriting system is terminating.\nWe define the following norm which assigns natural numbers to terms:\n\\[\n\\begin{array}{lll}\n\\norm{b} & = & 1\\\\\n\\norm{\\OR(X_1, \\ldots, X_k)} & = &\n \\norm{X_1} + \\ldots + \\norm{X_k} + 2\\\\\n\\norm{\\AND(X_1, \\ldots, X_k)} & = &\n 2\\cdot \\norm{X_1} \\cdot \\ldots \\cdot \\norm{X_k}\\\\\n\\norm{\\SAND(X_1, \\ldots, X_k)} & = &\n 2\\cdot \\norm{X_1} \\cdot \\ldots \\cdot \\norm{X_k}\\\\\n\\end{array}\n\\]\n\nIt can be easily verified that for every rewrite rule $l\\rightarrow r\\in \\thetrs$,\nwe have $\\norm{l} > \\norm{r}$. Consequently, there are no\ninfinite reduction sequences, or, in other words, the term rewriting\nsystem is strongly terminating.\nNotice that because we consider term rewriting modulo commutativity of\n$\\OR$ and $\\AND$, we have to verify that the left-hand side and the right-hand\nside of equations \\eqref{E1} and \\eqref{E2}\nhave equal norms~\\cite{KBV2001}. This is clearly the\ncase.\n\\item \nNow we prove that the terms \nproduced by the grammar~\\eqref{grammar:NF} \nare exactly the normal forms with respect to $\\thetrs$.\nFor the terms in $\\asterms$, none of the rewrite rules can be applied,\nbecause these terms do not contain $\\OR$, have no occurrences of\n$\\AND$ containing an argument of type $\\AND$, have no occurrences of\n$\\SAND$ containing an argument of type $\\SAND$, and do not contain\noperators with a single argument.\nWe extend this to terms $\\nfterms$ by observing that all $\\OR$\noperators occurring in such terms have at least two arguments and that\nall these arguments are different.\n\nConversely, consider a term $t$ in normal form that \ncontains an $\\OR$ operator. Then $t=\\OR(t_1,\\ldots,t_n)$, where the $t_i$\ndo not contain an $\\OR$ operator, else \\eqref{E3}, \\eqref{E10}, or\n\\eqref{E10'} can be applied. \nIt remains to show that normal form terms without occurrence of an\n$\\OR$ operator are in $\\asterms$. Such terms are basic terms or have\n$\\SAND$ or $\\AND$ as their top-level operator. The last two cases are\nsymmetric and we therefore only consider the case\n$\\AND(t_1,\\ldots,t_n)$. We must show that each $t_i$ is a basic term\nor in the form $t_i=\\SAND(t_1',\\ldots,t_m')$. Suppose not, then there\nexists a $t_i$ that has $\\AND$ as its top-level operator. It follows\nthat the term is not in normal form because \\eqref{E4} can be applied.\n\n\\item The normal forms are unique. \nTo show that the normal forms are unique, assume that \n$\\nfterm_1$ and $\\nfterm_2$ are both normal forms for a\n \\SAND~attack tree $t$. \nSince the rewrite system $\\thetrs$ \nwas constructed by orienting the axioms from $\\ESP$, \nwe have that $\\ESP\\vdash \\nfterm_1=\\nfterm_2$. \nThis means that $\\sem{\\nfterm_1}=\\sem{\\nfterm_2}$.\nFrom bijectivity proven in Lemma~\\ref{lem:bijection}, we obtain \n$\\nfterm_1 = \\nfterm_2$.\n\n\\item Now that we have proven termination and uniqueness of normal forms, it\nimmediately follows that the term rewriting system is\nconfluent~\\cite{D2005}.\n\\end{enumerate} \n\\qed\n\\end{proof}\n \n\n\n\n\nExample~\\ref{ex:nf} illustrates the notion of \ncanonical form for \\SAND~attack trees.\n}\n\n\\begin{example}\n\\label{ex:nf}\nThe canonical form of the \\SAND~attack tree $t$ \nin Figure~\\ref{fig:attack-tree} is the tree \n\\[\nt' = \\OR\\Big(\\SAND\\big(\\ftprhostsx, \\rshx, \\localbofx\\big), \\\\\n \\AND\\big(\\sshbofx, \\rsarefbofx\\big)\\Big)\n\\]\nshown in Figure~\\ref{fig:attack-tree2}. \nIt \nis easily seen to be in normal form with respect to $\\thetrs$.\n\\end{example}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[height=3.4cm]{atree3}\n\\caption{\\SAND\\ attack tree $t'$ equivalent to \\SAND\\ attack tree $t$ from \nFigure~\\ref{fig:attack-tree}}\n\\label{fig:attack-tree2}\n\\end{figure}\n\n\\subsection{SP Semantics as a Generalization of the Multiset Semantics}\n\\label{sec:SP_vs_M}\n\nHaving a complete set of axioms for the SP semantics\nallows us to formalize the relation between \n\\SAND\\ attack trees under the SP semantics and \nattack trees under the multiset semantics, denoted by $\\msem{\\cdot}$. \nThis is achieved by extracting \na complete set of axioms for \nthe multiset semantics for \nattack trees from \nthe set $\\ESP$. \nLet $\\ESM$ be the subset of axioms from $\\ESP$ that\ndo not contain the $\\SAND$ operator, \\ie, \n$\\ESM=\\{$\\eqref{E1}, \\eqref{E2}, \\eqref{E3}, \\eqref{E4}, \\eqref{E5},\n\\eqref{E6}, \\eqref{E10}, \\eqref{E11}$\\}$.\n\\begin{theorem}\n\\label{th:multiset-axioms}\nThe axiom system $\\ESM$\nis a complete set of axioms for the multiset\nsemantics for attack trees.\n\\end{theorem}\n{\n\\begin{proof}\nIn \\cite[Theorem 4.9]{KoMaRaSc_JLC}, a complete axiomatization of\nthe multiset semantics for an extention of attack trees \ncalled attack--defense trees (ADTrees) is given. In the following, we\ncall that axiomatization $\\EADT$.\nADTrees are a superset of attack trees.\nThey may contain defender's nodes \nmodeled by the so called\nopponent's functions and countermeasures. \nWe claim that $\\ESM$\nis a complete axiomatization of the multiset semantics for attack trees.\nObviously if two attack trees\nare equal with respect to $\\ESM$, then they\nare also equal with respect to $\\EADT$.\nThis is clear, because $\\ESM\\subset \\EADT$.\n\nConversely, we prove that if two attack trees are equal with respect to \n$\\EADT$, they are equal with respect to $\\ESM$. This\nfollows from the following syntactical reasoning. \n$\\EADT$ contains function symbols which we call\n\\emph{countermeasures}. \nObserve by inspecting the axioms of $\\EADT$ that if a\ncountermeasure occurs at the left-hand side of an equation, then it also\noccurs at the right-hand side, and vice versa. Therefore, axioms \n$(E_{13}),\n(E_{16}), (E_{17}), (E_{18}), (E_{19}), (E_{20})$ from $\\EADT$\ncan never be used in a derivation of equality of\ntwo standard attack trees. Further, observe that the remaining \naxioms $(E_9)$ and $(E_{12})$ from $\\EADT$ make use of \n\\emph{opponent's functions}.\nIn these axioms, \nan opponent function occurs on the left-hand side if and only\nif it occurs on the right hand side. Thus \nthese axioms are never used to equate\ntwo attack trees which do not contain opponent's nodes. \nThe remaining axioms are precisely \n\\eqref{E1}, \\eqref{E2}, \\eqref{E3}, \\eqref{E4}, \\eqref{E5},\n\\eqref{E6}, \\eqref{E10}, \\eqref{E11}.\nSo, we can only use these axioms \nto derive equalities\nof attack trees with respect to $\\EADT$, which implies that such a\nderivation is also possible using axioms from $\\ESM$.\n\\qed\n\\end{proof}\n}\n\n\nBy comparing the complete sets of axioms \n$\\ESP$ and $\\ESM$ we obtain that two attack trees are equivalent under\nthe multiset semantics if and only if they are equivalent under the SP \nsemantics. This is formalized in the following theorem. \n\\begin{theorem}\n\\label{th:SP_vs_M}\n$\\SAND$ attack trees under the SP semantics are a \\emph{conservative\nextension} of attack trees under the multiset semantics.\n\\end{theorem}\n\\begin{proof}\n{\nLet $t$ and $t'$ be standard attack trees.\nLet $\\msem{t}$ and $\\msem{t'}$ be their interpretation in the \nmultiset semantics and $\\sem{t}$ and\n$\\sem{t'}$ be their interpretation in the SP semantics.\nWe prove that $\\msem{t} = \\msem{t'}$ if and only if $\\sem{t} =\n\\sem{t'}$.\n\nBy Theorem~\\ref{th:multiset-axioms}, a complete axiomatization of \nthe multiset semantics for attack \ntrees \nconsists of axioms~\\eqref{E1}, \\eqref{E2}, \\eqref{E3}, \\eqref\n{E4}, \\eqref{E5}, \\eqref{E6}, \\eqref{E10}, \\eqref{E11}.\nThe complete axiomatization of the SP semantic for \n$\\SAND$ attack trees additionally \ncontains axioms \\eqref{E4'}, \\eqref{E6'},\nand \\eqref{E10'}. Thus, every equivalence of attack\ntrees under the multiset semantics \nis clearly an equivalence of $\\SAND$ attack trees\nunder the SP semantics.\n\nTo see the converse, \nwe show that the additional axioms do not introduce new\nequalities on standard attack trees. \nFirst inspect the three additional axioms and note that all of them\ncontain the $\\SAND$ operator.\n\nNext, observe that for all axioms, the set of variables occurring on the\nleft-hand side is equal to the set of variables occurring on the\nright-hand side. Thus, there is no axiom eliminating all\noccurrences of a variable. \nIn particular, we claim that \nall axioms transform \nterms containing a $p$-ary $\\SAND$ expression, where $p\\geq 2$, into\nterms containing a $q$-ary $\\SAND$ expression, for some $q\\geq 2$. \nThis is evident for equations without the $\\SAND$ operator (since no\nvariables are eliminated) and remains\nto be shown for equations \\eqref{E4'}, \\eqref{E6'},\nand \\eqref{E10'}. \nAxiom~\\eqref{E6'} introduces and removes unary $\\SAND$, but does not\nmodify the single variable $A$ and therefore satisfies the claim. \nThe arities of the two left-hand side $\\SAND$ operators \nin equation~\\eqref{E4'} are $l$ and $k+l+m$ and the arity of the \nright-hand side operator is $k+l+m$, where $k,m\\geq 0$ and $l\\geq 1$.\nSince $1\\leq l\\leq k+l+m$ and both sides contain a $\\SAND$ operator of\narity $k+l+m$, if either of the two sides\ncontains a $\\SAND$ operator with two or more arguments, then so does\nthe other side.\nFinally, since $l\\geq 1$, \nthe arity of the $\\SAND$ operator on the left-hand side of\nequation~\\eqref{E10'} is equal to the arities of the $\\SAND$ operators\non its right-hand side and at least one $\\SAND$ operator\noccurs on the right-hand side.\n\nWe can now show that none of the three axioms~\\eqref{E4'}, \\eqref{E6'}, \\eqref\n{E10'} introduces new equalities on\nstandard attack trees. \nIn particular, axiom~\\eqref{E6'} introduces and removes unary\n$\\SAND$, but this does not introduce new equalities on standard attack\ntrees. Equations~\\eqref{E4'} and~\\eqref{E10'} match unary $\\SAND$, but\nrequire a further $\\SAND$ with $2$ or more arguments to add a new equality.\nSince, by the above claim, no $p$-ary $\\SAND$ for $p\\geq 2$ \ncan be introduced with any of\nthe equations, the additional equations do not introduce new\nequalities on standard attack trees.\n\\qed\n}\n\n\\end{proof}\n\n{\n}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Attributes}\n\\label{sec:attributes}\n\n\nAttack trees do not only serve to represent security scenarios \nin a graphical way. They can also be used to \nquantify such scenarios with respect to \na given parameter, called an \\emph{attribute}. \nTypical examples of attributes include\nthe likelihood that the attacker's goal is satisfied and \nthe minimal time or cost of an attack. \nSchneier described~\\cite{Schn} \nan intuitive bottom-up algorithm for calculating attribute values\non attack trees:\nattribute values are assigned to the leaf nodes \nand two functions\\footnote{These are actually \nfamilies of functions representing infinitely many $k$-ary function symbols,\nfor all $k \\geq 2$.} (one for the \\OR\\ and one for the \\AND\\ refinement)\nare used to propagate the attribute value up to the root node. \nMauw and Oostdijk showed~\\cite{MaOo} that \nif the binary operations induced by the two functions \ndefine a semiring, \nthen the \nevaluation of the attribute on two attack trees \nequivalent with respect to the multiset semantics \nyields the same value. This result has \nbeen generalized to any semantics and attribute that satisfy a notion of \n\\emph{compatibility}~\\cite{KoMaRaSc_JLC}. \nWe briefly discuss it for \\SAND~attack trees \nat the end of this section. \nWe start with a demonstration on how \nthe bottom-up evaluation \nalgorithm \ncan naturally be extended to \\SAND\\ attack trees.\n \n \nAn {\\em attribute domain for an attribute $\\fullattr$ on \\SAND\\ attack trees} \nis a tuple $\\attrdomain_\\attr = (\\attrval_\\attr, \\attror_\\attr, \\attrand_\\attr, \\attrsand_\\attr)$\nwhere $\\attrval_\\attr$ is a set of values and \n $\\attror_\\attr, \\attrand_\\attr, \\attrsand_\\attr$ are families of \n$k$-ary functions of the\n form $\\attrval_\\attr\\times\\dots\\times\\attrval_\\attr\\to\\attrval_\\attr$, \nassociated to \\OR, \\AND, and \\SAND\\ refinements, respectively. \nAn {\\em attribute for \\SAND~attack trees} is a pair \n$\\fullattr = (\\attrdomain_\\attr, \\basicassign_\\attr)$ \nformed by an attribute domain $\\attrdomain_\\attr$ and a function \n$\\basicassign_\\attr:\\basicact\\to\\attrval_\\attr$,\ncalled {\\em basic assignment} for $\\fullattr$, which associates a value from \n$\\attrval_\\attr$ with each basic action $b \\in \\basicact$. \n\\begin{definition}\n\\label{def:attr}\nLet $\\fullattr = \\big((\\attrval_\\attr, \\attror_\\attr, \\attrand_\\attr, \\attrsand_\n\\attr), \\basicassign_\\attr\\big)$\nbe an attribute. \nThe attribute evaluation function \n$\\attr: \\sandtree \\to \\attrval_\\attr$ which calculates the value of \nattribute \n$\\fullattr$ \nfor every \\SAND \\ attack tree $t \\in \\sandtree$\nis defined recursively as follows\n\\[ \\attr(t) = \\left\\{ \n \\begin{array}{l l}\n \\basicassign_\\attr(t) & \\quad \\text{if $t=b,\\ b\\in\\basicact$}\\\\\n \\attror_\\attr\\big(\\attr(t_1), \\dots, \\attr(t_k)\\big) & \\quad \\text{if $t=\\OR(t_1, \\dots, t_k)$}\\\\\n \\attrand_\\attr\\big(\\attr(t_1), \\dots, \\attr(t_k)\\big) & \\quad \\text{if $t=\\AND(t_1, \\dots, t_k)$}\\\\\n \\attrsand_\\attr\\big(\\attr(t_1), \\dots, \\attr(t_k)\\big) & \\quad \\text{if $t=\\SAND(t_1, \\dots, t_k)$}\n \\end{array} \\right.\\]\n\\end{definition}\n\n\nThe following example illustrates the bottom-up evaluation\nof the attribute \\emph{minimal attack time} on the \\SAND~attack trees \ngiven in Example~\\ref{eg:def-attack-tree}.\n\\begin{example}\n\\label{ex:min_cost}\nLet $\\attr$ denote the minimal time \nthat the attacker needs to achieve her goal.\nWe make the following assignments to the basic actions: \n$\\ftprhostsx\\mapsto 3$, $\\rshx\\mapsto 5$, $\\localbofx\\mapsto 7$, $\\sshbofx\\mapsto \n8$, $\\rsarefbofx\\mapsto 9$. \nSince we are interested in the minimal attack time, \nthe function for an \\OR\\ node \nis defined by \n$\\attror_\\attr(x_1,\\dots,x_k)=\\min\\{x_1,\\dots, x_k\\}$. \nThe function for an \\AND\\ node \nis $\\attrand_\\attr(x_1,\\dots,x_k)=\\max\\{x_1,\\dots, x_k\\}$, \nwhich models that the children \nof a conjunctively refined node are executed in parallel.\nFinally, in order to model that the children \nof a \\SAND\\ node need to be executed \nsequentially, \nwe let \n$\\attrsand_\\attr(x_1,\\dots,x_k)=\\sum_{i=1}^{k} x_i$. \nAccording to Definition~\\ref{def:attr}, \nthe minimal attack time for our running scenario $t$ is\n\\[\\attror_\\attr\\Big(\\attrsand_\\attr\\big(\\attrsand_\\attr(3,5),7\\big),\n\\attrand_\\attr(8, 9)\\Big)=\n\\min\\Big({\\mathrm\\Sigma}\\big({\\mathrm\\Sigma}(3,5),7\\big), \\max(8, 9)\\Big)=9.\\]\n\\end{example}\n\nIn the case of standard attack trees, \nthe bottom-up procedure uses only two functions to propagate \nthe attribute values to the root -- one for conjunctive and one for \ndisjunctive nodes. This means that the same function is employed \nto calculate the value of every conjunctively refined node, \nindependently of whether its children need to be executed sequentially or can \nbe executed simultaneously. \nEvidently, with \\SAND~attack trees, we can apply different propagation \nfunctions for \\AND\\ and \\SAND\\ nodes, as in \nExample~\\ref{ex:min_cost}. \nTherefore, \\SAND~attack trees can be evaluated over a larger set of \nattributes, and hence may provide more accurate evaluations of \nattack \nscenarios than \nstandard attack trees.\n\n\n\nTo guarantee that the evaluation of an attribute on equivalent \nattack trees yields\nthe same value, \nthe attribute domain must be \n\\emph{compatible} with a considered semantics~\\cite{KoMaRaSc_JLC}. \nOur complete set of axioms is a useful tool \nto check for compatibility. \nConsider an attribute domain\n$\\attrdomain_\\attr = (\\attrval_\\attr, \\attror_\\attr, \\attrand_\\attr, \\attrsand_\\attr)$, \nand let $\\sigma$ be a mapping\n$\\sigma=\\{\\OR\\mapsto \\attror_\\attr, \\AND \\mapsto \\attrand_\\attr, \n\\SAND \\mapsto \\attrsand_\\attr\\}$. \nGuaranteeing that $\\attrdomain_\\attr$ is compatible with \na semantics axiomatized by $E$ \namounts to verifying that the equality \n$\\sigma(l)=\\sigma(r)$ holds in $\\attrval_\\attr$, for every axiom\n$l=r\\in E$.\nIt is an easy exercise to show that the attribute domain \nfor minimal attack time, considered in Example~\\ref{ex:min_cost},\nis compatible with the SP semantics for \\SAND\\ attack trees.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have formalized the extension of attack trees \nwith sequential conjunctive refinement, \ncalled \\SAND, \nand given a semantics to \\SAND~attack trees in terms of\nsets of series-parallel graphs.\nThis SP semantics \nnaturally extends the multiset semantics \nfor attack trees from~\\cite{MaOo}.\nWe have shown that the notion of \na complete set of axioms for a semantics and \nthe bottom-up evaluation procedure can be generalized \nfrom attack trees to \\SAND~attack trees, \nand have proposed a complete axiomatization of the SP semantics. \n \n\nA number of recently proposed solutions\nfocus on extending attack trees with defensive\nmeasures~\\cite{RoKiTr2,KoMaRaSc_JLC}.\nThese extensions support reasoning about security scenarios\ninvolving two players -- an attacker and a defender --\nand the interaction between them.\nIn future work, we intend to add\nthe $\\SAND$ refinement to such trees.\nAfterwards, we plan to investigate\nsequential disjunctive refinement, as used\nfor instance in~\\cite{Arnold-POST14}. Our goal is to propose\na complete formalization\nof trees with attack and defense nodes,\nthat have parallel and sequential, conjunctive and disjunctive\nrefinements. \nThe findings will be implemented in the software \napplication ADTool~\\cite{adtool}. \n\n\n\n\\paragraph{\\textbf{Acknowledgments}} The research leading\nto these results has received funding from the European Union Seventh Framework\nProgramme under grant agreement number 318003 (TREsPASS) and from\nthe Fonds National de la Recherche Luxembourg under grant \nC13\/IS\/5809105.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}