\documentclass[12pt]{amsart} %\usepackage{amsfonts, amsmath, amstext, latexsym} \usepackage{graphpap} %\usepackage[all]{xy} %\CompileMatrices \setlength{\textwidth}{16cm} \setlength{\oddsidemargin}{0.5cm} \setlength{\evensidemargin}{0.5cm} \begin{document} \newcommand{\ii}{\hookrightarrow} \newcommand{\hh}{\ensuremath{\mathbb H}} \newcommand{\ee}{\ensuremath{\mathbb E}} \newcommand{\rr}{\ensuremath{\mathbb R}} \newcommand{\qq}{\ensuremath{\mathbb Q}} \newcommand{\zz}{\ensuremath{\mathbb Z}} \newcommand{\cc}{\ensuremath{\mathcal{C}}} \newcommand{\ff}{\ensuremath{\mathcal{F}}} \newcommand{\uu}{\ensuremath{\mathcal{U}}} \newcommand{\al}{\ensuremath{\alpha}} \newcommand{\bb}{\ensuremath{\beta}} \newcommand{\be}{\ensuremath{\beta}} \newcommand{\g}{\ensuremath{\gamma}} \newcommand{\G}{\ensuremath{\Gamma}} \newcommand{\de}{\ensuremath{\delta}} \newcommand{\D}{\ensuremath{\Delta}} \newcommand{\ep}{\ensuremath{\varepsilon}} \newcommand{\f}{\ensuremath{\varphi}} \newcommand{\s}{\ensuremath{\sigma}} \newcommand{\la}{\ensuremath{\lambda}} \newcommand{\w}{\ensuremath{\omega}} \newcommand{\W}{\ensuremath{\Omega}} \newcommand{\p}{\ensuremath{\partial}} \newcommand{\M}{\ensuremath{\widetilde{M}}} \newcommand{\e}{\ensuremath{\epsilon}} \newcommand{\pp}{\ensuremath{\parallel}} \newcommand{\inv}{\ensuremath{^{-1}}} %\newcommand{\qed}{$\Box$} \newcommand{\se}{\ensuremath{\subseteq}} \newcommand{\lra}{\ensuremath{\longrightarrow}} \newcommand{\lla}{\ensuremath{\longleftarrow}} \newcommand{\pf}{{\noindent\bf Proof.\ }} \newcommand{\noi}{\noindent} \newcommand{\ds}{\displaystyle} %\newcommand{\xo}{\ensuremath{X^{(0)}}} \newcommand{\go}{\ensuremath{\G^{(0)}}} \newcommand{\fff}{\ensuremath{\bar{f}}} \newcommand{\?}{{\bf ?????}} \newcommand{\bu}{$\bullet$\ } \newtheorem{ttt}{Theorem} \newtheorem{lll}[ttt]{Lemma} \newtheorem{slll}[ttt]{Sublemma} \newtheorem{ddd}[ttt]{Definition} \newtheorem{ppp}[ttt]{Proposition} \newtheorem{ccc}[ttt]{Corollary} \newtheorem{nnn}[ttt]{Not a theorem} \newtheorem{ttt-in-text}{Theorem} %\renewcommand{\baselinestretch}{1.3} %\small\normalsize \title[The Baum-Connes conjecture for hyperbolic groups] {The Baum-Connes conjecture for hyperbolic groups} \maketitle \centerline{Igor Mineyev and Guoliang Yu\footnote{ The second author is partially supported by NSF and MSRI. }} \date{} \begin{abstract} {We prove the Baum-Connes conjecture for hyperbolic groups and \\ their subgroups. } \end{abstract} %\maketitle %\renewcommand{\thefootnote}{\fnsymbol{footnote}} %\footnote{\textit{Date}: The second author is %partially supported by %NSF and MSRI } %\footnote[0]{1991 \textit{Mathematics Subject Classification}: %} %\setcounter{section}{-1} \section{Introduction.} The Baum-Connes conjecture states that, for a discrete group $G$, the K-homology groups of the classifying space for proper $G$-action is isomorphic to the K-groups of the reduced group $C^{\ast}$-algebra of $G$~\cite{BC, BCH}. A positive answer to the Baum-Connes conjecture would provide a complete solution to the problem of computing higher indices of elliptic operators on compact manifolds. The rational injectivity part of the Baum-Connes conjecture implies the Novikov conjecture on homotopy invariance of higher signatures. The Baum-Connes conjecture also implies the Kadison-Kaplansky conjecture that for $G$ torsion free there exists no non-trivial projection in the reduced group $C^{\ast}$-algebra associated to $G$. In~\cite{HK}, Higson and Kasparov prove the Baum-Connes conjecture for groups acting properly and isometrically on a Hilbert space. In a recent remarkable work, Vincent Lafforgue proves the Baum-Connes conjecture for strongly bolic groups with property RD~\cite{L1, L2, L3}. In particular, this implies the Baum-Connes conjecture for the fundamental groups of strictly negatively curved compact manifolds. In~\cite{CM}, Connes and Moscovici prove the rational injectivity part of the Baum-Connes conjecture for hyperbolic groups using cyclic cohomology method. In this paper, we exploit Lafforgue's work to prove the Baum-Connes conjecture for hyperbolic groups and their subgroups. The main step in the proof is the following theorem. \setcounter{ttt-in-text}{\ref{metric}} \addtocounter{ttt-in-text}{-1} \begin{ttt-in-text} Every hyperbolic group $G$ admits a metric $\hat{d}$ with the following properties. \begin{itemize} \item [(1)] $\hat{d}$ is $G$-invariant, i.e. $\hat{d}(g\cdot x,g\cdot y)=\hat{d}(x,y)$ for all $x,y,g\in G$. \item [(2)] $\hat{d}$ is quasiisometric to the word metric. \item [(3)] The metric space $(G,\hat{d})$ is weakly geodesic and strongly bolic. \end{itemize} \end{ttt-in-text} This paper is organized as follows. In section 2, we recall the concepts of hyperbolic groups and bicombings. In section 3, we introduce a distance-like function $r$ on a hyperbolic group and study its basic properties. In section 4, we prove that $r$ satisfies certain distance-like inequalities. In section 5, we construct a metric $\hat{d}$ on a hyperbolic group and prove Theorem~\ref{metric} stated above. In section 6, we combine Lafforgue's work and Theorem 17 to prove the Baum-Connes conjecture for hyperbolic groups and their subgroups. After this work was done, we learned from Vincent Lafforgue that he has independently proved the Baum-Connes conjecture for hyperbolic groups by a different and elegant method~\cite{L4}, and we also learned from Michael Puschnigg that he has independently proved the Kadison-Kaplansky conjecture for hyperbolic groups using a beautiful local cyclic homology method~\cite{p}. It is our pleasure to thank both of them for bringing their work to our attention. \section{Hyperbolic groups and bicombings.} In this section, we recall the concepts of hyperbolic groups and bicombings. \subsection{Hyperbolic groups.} \label{s_hyp} Let $G$ be a finitely generated group. Let $S$ be a finite generating set for $G$. Recall that the Cayley graph of $G$ with respect to $S$ is the graph $\G$ satisfying the following conditions: \begin{itemize} \item [(1)] the set of vertices in $\G$, denoted by $\go$, is $G$; \item [(2)] the set of edges is $G\times S$, where each edge $(g,s)\in G\times S$ spans the vertices $g$ and $gs.$ \end{itemize} We endow $\G$ with the {\sf path metric}~$d$ induced by assigning length 1 to each edge. Notice that $G$ acts freely, isometrically and cocompactly on $\G$. A {\sf geodesic path} in $\G$ is a shortest edge path. The restriction of the path metric $d$ to $G$ is called the {\sf word metric}. A finitely generated group $G$ is called {\sf hyperbolic}\, if there exists a constant $\de\ge 0$ such that all the geodesic triangles in $\G$ are {\sf $\de$-fine} in the following sense: if $a$, $b$, and $c$ are vertices in $\G$, $[a,b]$, $[b,c]$, and $[c,a]$ are geodesics from $a$ to $b$, from $b$ to $c$, and from $c$ to $a$, respectively, and points $\bar{a}\in [b,c]$, $v,\bar{c}\in [a,b]$, $w,\bar{b}\in [a,c]$ satisfy $$d(b,\bar{c})=d(b,\bar{a}),\quad d(c,\bar{a})=d(c,\bar{b}),\quad d(a,v)=d(a,w)\le d(a,\bar{c})=d(a,\bar{b}),$$ then $d(v,w)\le \de$. The above definition of hyperbolicity does not depend on the choice of the finite generating set $S$. See~\cite{G, ABC} for other equivalent definitions. For vertices $a$, $b$, and $c$ in $\G$, the {\sf Gromov product} is defined by $$(b|c)_a := d(a,\bar{b})= d(a,\bar{c})= \frac{1}{2}\Big[d(a,b)+d(a,c)-d(b,c)\Big].$$ The Gromov product can be used to measure the degree of cancellation in the multiplication of group elements in $G$. \subsection{Bicombings.} Let $G$ be a finitely generated group. Let $\G$ be a Cayley graph with respect to a finite generating set. A {\sf bicombing} $p$ in $\G$ is a function assigning to each ordered pair $(a,b)$ of vertices in $\G$ an oriented edge-path $p[a,b]$ from $a$ to $b$. A bicombing $p$ is called {\sf geodesic}\, if each path $p[a,b]$ is geodesic, i.e. a shortest edge path. A bicombing $p$ is {\sf $G$-equivariant} if $p[g\cdot a, g\cdot b]= g\cdot p[a,b]$ for each $a,b\in\go$ and each $g\in G$. %\input{bch-i.txt} \section{Definition and properties of $r(a,b)$.} The purpose of this section is to introduce a distance-like function $r$ on a hyperbolic group and study its basic properties. Let $G$ be a hyperbolic group and $\G$ be a Cayley graph of $G$ with respect to a finite generating set. We endow $\G$ with the path metric $d$, and identify $G$ with $\go$, the set of vertices of $\G$. Let $\de\ge 1$ be a positive integer such that all the geodesic triangles in $\G$ are $\de$-fine. The {\sf ball} $B(x,R)$ is the set of all vertices at distance at most $R$ from the vertex $x$. The {\sf sphere} $S(x,R)$ is the set of all vertices at distance $R$ from the vertex $x$. Pick an equivariant geodesic bicombing~$p$ in~$\G$. By $p[a,b](t)$ we denote the point on the geodesic path $p[a,b]$ at distance $t$ from~$a$. Recall that $C_0 (G,\qq)$ is the space of all $0$-chains (in $G=\go$) with coefficients in $\qq$. Endow $C_0(G,\qq)$ with the $\ell^1$-norm $|\cdot|_1$. We identify $G$ with the standard basis of $C_0(G,\qq)$. Therefore the left action of $G$ on itself induces a left action on~$C_0(G,\qq)$. First we recall several constructions from~\cite{M3}. For $v, w\in G$, the {\sf flower} at $w$ with respect to $v$ is defined to be $$ Fl(v, w) := S(v, d(v,w))\cap B(w,\de)\subseteq G.$$ For each $a\in G$, we define $pr_a: G\rightarrow G$ by: \begin{itemize} \item [(1)] $pr_a (a) := a;$ \item [(2)] if $b\neq a$, $pr_a(b) := p[a,b](t)$, where $t$ is the largest integral multiple of $10\de$ which is strictly less than $d(a,b)$. \end{itemize} Now for each pair $a,b\in G$, we define a $0$-chain $f(a,b)$ in $G$ inductively on the distance $d(a,b)$ as follows: \begin{itemize} \item [(1)] if $d(a,b)\leq 10\de$, $f(a,b) :=b$; \item [(2)] if $d(a,b) >10\de$ and $d(a, b)$ is not an integral multiple of $10\de$, let $f(a,b) := f(a, pr_a (b));$ \item [(3)] if $d(a,b)> 10\de$ and $d(a,b)$ is an integral multiple of $10\de$, let $$ f(a,b) := \frac{1}{\# Fl(a,b)}\sum_{x\in Fl(a,b)} f(a, pr_a (x)).$$ \end{itemize} \begin{ppp}[\cite{M3}] \label{c-c-c} The function $f:G\times G\to C_0(G,\qq)$ defined above satisfies the following conditions. \begin{itemize} \item [(1)] For each $a,b\in G$, $f(b,a)$ is a {\sf convex combination}, i.e. its coefficients are non-negative and sum up to~1. \item [(2)] If $d(a,b)\ge 10\de$, then $supp\,f(b,a)\subseteq B(p[b,a](10\de),\de)\cap S(b,10\de)$. \item [(3)] If $d(a,b)\le 10\de$, then $f(b,a)=a$. \item [(4)] $f$ is $G$-equivariant, i.e. $f(g\cdot b, g\cdot a)= g\cdot f(b,a)$ for any $g,a,b\in G$. \item [(5)] There exist constants $L\ge 0$ and $0\le\la<1$ such that, for all $a,a',b\in G$, $$\Big|f(b,a)-f(b,a')\Big|_1\le L\,\la^{(a|a')_b}.$$ \end{itemize} \end{ppp} \setlength{\unitlength}{.7cm} \begin{figure}[h] \begin{center} \begin{picture}(12,5.2) %\input{fig/sh_defn-f} \put(1,2.5){\circle*{0.15}} \put(.3,2.4){\small$a$} \put(11,2.5){\circle*{0.15}} \put(11.4,2.35){\small$b$} \put(5,2.9){\footnotesize$p[b,a]$} \put(9.3,2.6){\tiny $10\de$} %from a to b \qbezier(1,2.5)(6,2.5)(11,2.5) %f(a,b) \qbezier(8.5,2)(8.3,2.5)(8.5,3) \put(7.8,1.4){\footnotesize$f(b,a)$} \put(8.41,2.5){\circle*{0.15}} \end{picture} \end{center} \caption{Convex combination $f(b,a)$.} \end{figure} Let $\w_7$ be the number of elements in a ball of radius $7\de$ in $G$. For each $a\in G$, a 0-chain $star(a)$ is defined by $$star(a):=\frac{1}{\w_7}\sum_{x\in B(a,7\de)} x.$$ This extends to a linear operator $star: C_0(G,\qq)\to C_0(G,\qq)$. Define the 0-chain $\fff(b,a)$ by $\fff(b,a):=star\big( f(b,a) \big)$. The main reason for introducing $\fff$ is that $\fff$ has better cancellation properties than $f$ (compare Proposition~\ref{c-c-c}(5) with Proposition~\ref{la'}(5) and~\ref{la'}(6) below). These cancellation properties play key roles in this paper. \begin{ppp}[\cite{M3}] \label{la'} The function $\fff:G\times G\to C_{0}(G,\qq)$ defined above satisfies the following conditions. \begin{itemize} \item [(1)] For each $a,b\in G$, $\fff(b,a)$ is a convex combination. \item [(2)] If $d(a,b)\ge 10\de$, then $supp\,\fff(b,a)\subseteq B(p[b,a](10\de),8\de)$. \item [(3)] If $d(a,b)\leq 10\de$, then $supp\, \fff(b,a) \subseteq B(a, 7\de).$ \item [(4)] $\fff$ is $G$-equivariant, i.e. $\fff(g\cdot b, g\cdot a)= g\cdot \fff(b,a)$ for any $g,a,b\in G$. \item [(5)] There exist constants $L\ge 0$ and $0\le\la<1$ such that, for all $a,a',b\in G$, $$\Big|\fff(b,a)-\fff(b,a')\Big|_1\le L\,\la^{(a|a')_b}.$$ \item [(6)] There exists a constant $0\le\la'<1$ such that if $a,b,b'\in G$ satisfy $(a|b)_{b'}\le 10\de$ and $(a|b')_b\le 10\de$, then $\Big|\fff(b,a)-\fff(b',a)\Big|_1\le 2\la'$. \item [(7)] Let $a,b,c\in G$, $\gamma$ be a geodesic path from $a$ to $b$, and let $$c\in N_G(\gamma, 9\delta):=\{x\in G\ \big|\ d(x,\gamma)\leq 9\delta\}.$$ Then $supp (\bar{f}(c,a))\subseteq N_G (\gamma, 9\delta).$ \end{itemize} \end{ppp} \begin{ddd} \label{def-r} For each pair of vertices $a,b\in G$, a rational number $r(a,b)\ge 0$ is defined inductively on $d(a,b)$ as follows. \begin{itemize} \item $r(a,a):=0$. \item If $0< d(a,b)\le 10\de$, let $r(a,b):= 1$. \item If $d(a,b)> 10\de$, let $r(a,b):=r\big(a,\fff(b,a)\big)+ 1$, where $r\big(a,\fff(b,a)\big)$ is defined by linearity in the second variable. \end{itemize} \end{ddd} The function $r$ is well defined by Proposition~\ref{la'}(2). Also, $r(a,b)$ is well defined when $b$ is a 0-chain, by linearity. Let $\qq_{\ge 0}$ denote the set of all non-negative rational numbers. \begin{ppp} \label{N} For the function $r: G\times G\to \qq_{\ge 0}$ defined above, there exists $N\ge 0$ such that, for all $a,b,b'\in G$, $$\big|r(a,b)-r(a,b')\big|\le d(b,b')+N.$$ \end{ppp} \pf Up to the $G$-action, there are only finitely many triples of vertices $a$, $b$, $b'$, satisfying $d(a,b)+d(a,b')\le 40\de$, hence there exists a uniform bound $N'$ for the norms $$\big|r(a,b)-r(a,b')\big|$$ for such vertices $a$, $b$, $b'$. Let $\la'$ be the constant from Proposition~\ref{la'}(6) and pick $N$ large enough so that \begin{equation} \label{N-def} N'\le N\qquad\mbox{and}\qquad \la'\cdot[27\de+N]\le N. \end{equation} We shall prove the inequality in Proposition 4 by induction on $d(a,b)+d(a,b')$. If $d(a,b)+d(a,b')\le 40\de$, then $$\big|r(a,b)-r(a,b')\big|\le N'\le N\le d(b,b')+N$$ just by the choices of $N'$ and $N$. We assume now that $d(a,b)+d(a,b')> 40\de$. Consider the following two cases. \noi{\underline{\bf Case~1.} {\it \ $(a|b')_b> 10\de$\quad or\quad $(a|b)_{b'}> 10\de$.} \setlength{\unitlength}{1 cm} \begin{figure}[h] \begin{center} \begin{picture}(8,4.5) %\graphpaper[1](0,0)(8,4.5) \put(0.5,1.5){\circle*{0.1}} \put(0.1,1.5){\small$a$} \put(7,4){\circle*{0.1}} \put(7.2,4){\small$b$} \put(6,0){\circle*{0.1}} \put(6.2,-0.1){\small$b'$} %from a to b \qbezier(0.5,1.5)(4.5,2)(7,4) \put(2.8,2.3){\footnotesize$p[b,a]$} %from a to b' \qbezier(0.5,1.5)(4.5,2)(6,0) %from b to b' \qbezier(7,4)(4.5,2)(6,0) \put(6,1.1){\small$\g$} %inscribed triangle \put(4.42,2.49){\circle*{0.1}} \put(5.455,1.7){\circle*{0.1}} \put(4.47,1.18){\circle*{0.1}} \put(6.2,2.8){\tiny$10\de$} \put(5.5,1.97){\circle*{0.1}} \put(5.7,1.9){\small$v'$} \put(4.7,2.6){\circle*{0.1}} \put(5,2.48){\small$v$} \put(4.7,3.9){\circle*{0.1}} \put(4.2,3.9){\small$x$} %from v to x \qbezier(4.7,2.6)(4.7,3.25)(4.7,3.9) %from v to v' \qbezier(4.68,2.6)(5.09,2.285)(5.5,1.97) \end{picture} \end{center} \caption{\label{case1} Proposition~\ref{N}, Case~1.} \end{figure} Assume, for example, that $(a|b')_b> 10\de$. Then $d(a,b)> 10\de$, hence, by definition, $$r(a,b)=r\big(a,\fff(b,a)\big)+ 1.$$ By Proposition 2(2), we have $supp\,\fff(b,a)\se B(v,8\de)$, where $v:=p[b,a](10\de)$. Also, $(a|b')_b>10\de$ implies $d(b,b')> 10\de$. Hence there exists a geodesic $\g$ between $b$ and $b'$, and a vertex $v'$ on $\g$ with $d(b,v')=d(b,v)=10\de$. Since geodesic triangles are $\de$-fine, $d(v,v')\le\de$. For every $x\in supp\,\fff(b,a)$, \begin{eqnarray*} d(x,b')&&\le d(x,v)+d(v,v')+d(v',b')\\ &&\le 8\de+\de+\big[d(b,b')- 10\de\big]\\ &&\le d(b,b')-1,\\ d(a,x) && \le d(a,v)+d(v,x)\\ &&\le \big[d(a,b)-10\de\big]+ 8\de\\ && \le d(a,b)-1. \end{eqnarray*} Therefore $$d(a,x)+d(a,b') 40\de$ and $d(b,b')=(a|b')_b+(a|b)_{b'}\le 20\de$, we have $d(a,b)> 10\de$ and $d(a,b')> 10\de$. Then, by the definition of $r$, \begin{eqnarray} \label{cond3-3} &&\big| r(a,b)-r(a,b') \big|\\ &&= \Big| r\big(a,\fff(b,a)\big)+ 1- r\big(a,\fff(b',a)\big)- 1 \Big| \nonumber \\ &&= \Big|r\big(a,\fff(b,a)- \fff(b',a)\big)\Big|. \nonumber \end{eqnarray} The 0-chain $\fff(b,a)-\fff(b',a)$ can be represented in the form $f_+-f_-$, where $f_+$ and $f_-$ are 0-chains with non-negative coefficients and disjoint supports. By Proposition~\ref{la'}(6), \begin{eqnarray*} |f_+|_1+|f_-|_1&&= |f_+ - f_-|_1\\ &&=\big|\fff(b,a)-\fff(b',a)\big|_1 \\ &&\le 2\la'.\end{eqnarray*} Since the coefficients of the 0-chain $f_+ - f_-=\fff(b,a)-\fff(b',a)$ sum up to 0, then \begin{equation} \label{f+-} |f_+|_1=|f_-|_1\le\la'. \end{equation} With the notations $v:=p[b,a](10\de)$, $v':=p[b',a](10\de)$, we have \begin{eqnarray*} &&supp\,f_+\se supp\,\fff(b,a)\se B(v,8\de)\quad\mbox{and}\\ &&supp\,f_-\se supp\,\fff(b',a)\se B(v',8\de) \end{eqnarray*} (see Figure~\ref{case2}). Since geodesic triangles are $\delta$-fine, there exists a point $w$ on $p[b,a]$ such that $d(a,w)=d(a,v')$ and $d(w,v')\leq \delta.$ We first assume that $d(a,w)\leq d(a,v).$ We have \begin{eqnarray*} d(v,v') && \leq d(v,w)+ d(w,v')\\ &&\leq d(w, \overline{b'})+\delta \\ && = d(v', \overline{b})+\delta \\ &&\leq 11\delta, \end{eqnarray*} where $\overline{b'}$ and $\overline{b}$ are inscribed points as in the definition of $\delta$-fine triangle in section~\ref{s_hyp}. If $d(a,w)>d(a,v),$ we can apply the same argument to prove $d(v,v')\leq 11 \delta$ by interchanging $v'$ with $v$. Hence by Proposition 2(2), for each $x\in supp\,f_+$ and $x'\in supp\,f_-$, \begin{eqnarray*} d(x,x') && \le d(x,v)+d(v,v') +d(v', x')\\ && \le 8\de+ 11\de+ 8\de \\ &&= 27\de. \end{eqnarray*} Also $d(a,x)+d(a,x')< d(a,b)+d(a,b')$, so the induction hypotheses for the vertices $a$, $x$, and $x'$ apply, giving \begin{eqnarray} \label{ineq1} \big|r(a,x)-r(a,x')\big|&& \le d(x,x')+N\\ &&\le 27\de +N \nonumber \end{eqnarray} for each $x\in supp\,f_+$ and $x'\in supp\,f_-$. Then we continue equality~(\ref{cond3-3}) using (\ref{f+-}), (\ref{ineq1}), linearity of $r$ in the second variable, and the definition of $N$ in (\ref{N-def}): \begin{eqnarray*} \big|r(a,b)-r(a,b')\big|&& = \Big|r\big(a,\fff(b,a)-\fff(b',a)\big)\Big|\\ &&= \Big|r\big(a,f_+)-r(a,f_-)\Big|\\ &&\le \la'\cdot \big[27\de+ N\big]\\ &&\le N\le d(b,b')+N. \end{eqnarray*} Proposition~\ref{N} is proved.\qed Let $\ep: C_0(G,\qq)\to\qq$ be the augmentation map taking each 0-chain to the sum of its coefficients. A 0-chain~$z$ with $\ep(z)=0$ is called a {\sf 0-cycle}. \begin{ppp} \label{D} There exists a constant $D\ge 0$ such that, for each $a\in G$ and each 0-cycle~$z$, $$\big| r(a,z) \big|\le D\, |z|_1\, diam \big(supp(z)\big).$$ \end{ppp} \pf It suffices to consider the case $z=b-b'$, where $b$ and $b'$ are vertices with $d(b,b')=1$. But this case is immediate from Proposition~\ref{N} by taking $D:=\frac{1}{2}(1+N)$. \qed \begin{ttt} \label{r} For a hyperbolic group $G$, the function $r: G\times G\to\qq_{\ge 0}$ from Definition~\ref{def-r} satisfies the following properties. \begin{itemize} \item [(1)]$r$ is $G$-equivariant, i.e. $r(a,b)= r(g\cdot a, g\cdot b)$ for $g,a,b\in G$. \item [(2)] $r$ is Lipschitz equivalent to the distance function. More precisely, we have $$\ds\frac{1}{10\de}\,d(a,b)\le r(a,b)\le d(a,b)$$ for all $a,b\in G$. \item [(3)] There exist constants $C\ge 0$ and $0\le\mu<1$ such that, for all \mbox{$a, a', b, b'\in G$} with $d(a,a')\le 1$ and $d(b,b')\le 1$, $$\big| r(a,b)- r(a',b) - r(a,b')+ r(a',b') \big|\le C\mu^{d(a,b)}.$$ In particular, if $d(a,a')\leq 1$ and $d(b,b')\leq 1$, then $$\big| r(a,b)- r(a',b)- r(a,b')+ r(a',b') \big| \to 0\quad \mbox{as}\quad d(a,b)\to\infty.$$ \end{itemize} \end{ttt} \pf \boldmath $(1)$ \unboldmath The $G$-equivariance of $r$ follows from the definition of $r$ and Proposition~\ref{la'}(4).\\ \boldmath $(2)$ \unboldmath Using the assumption that $\delta\geq 1$ and the definition of $r$, the inequalities \begin{equation*} \frac{1}{10\de}d(a,b)\le r(a,b)\le d(a,b) \end{equation*} can be shown by an easy induction on $d(a,b)$. The remaining part \boldmath $(3)$ \unboldmath immediately follows from the following proposition. \begin{ppp} \label{AB-rho} There exist constants $A> 0$, $B> 0$, and $0< \rho <1$ such that, for all $a,a',b,b'\in G$ with $d(a,a')\le 1$ and $d(b,b')\le 30\de$, $$\big| r(a,b)- r(a',b)- r(a,b')+ r(a',b') \big|\le \big( A\ d(b,b')+ B\big)\,\rho^{d(a,b)+d(a,b')}.$$ \end{ppp} \pf Let $D\ge 0$ be the constant from Proposition~\ref{D}, $L\ge 0$ and $0\le\la <1$ be the constants from Propositions~\ref{c-c-c}(5) and~\ref{la'}(5), $\de\ge 1$ be an integral hyperbolicity (fine-triangles) constant, and $\w_7$ be the number of vertices in a ball of radius $7\de$ in $\G$. Now we define constants $A$, $B$ and $\rho$. Since the inequality obviously holds when $b=b'$, we will assume that $d(b,b')\ge 1$. Then constant $A>0$ can be chosen large enough so that \begin{itemize} \item the desired inequality is satisfied whenever $d(a,b)+d(a,b')\le 100\de$, $\rho\ge \sqrt{\la}$, and $B> 0$, and \item $32D\de L\big(\sqrt{\la}\big)^{-32\de}< A$. \end{itemize} So from now on we can assume that $d(a,b)+d(a,b')> 100\de$. Also the choice of $A$ implies that inequalities $$1-\frac{A}{Al+B} + \frac{32D\de L\big(\sqrt{\la}\big)^{t-32\de}}{(Al+B)\rho^{t-18\de}}\le 1-\frac{A}{Al+B}+ \frac{32D\de L\big(\sqrt{\la}\big)^{-32\de}}{Al+B}< 1$$ hold for all $B>0$, $\sqrt{\la}\le \rho<1 $, $1\le l\le 30\de$, and $t\ge 0$. Therefore, we can pick $B>0$ sufficiently large and $\rho <1$ sufficiently close to~1 so that the inequalities \begin{eqnarray} \nonumber && 1-\frac{A}{Al+B} + \frac{32D\de L\big(\sqrt{\la}\big)^{t-32\de}}{(Al+B)\rho^{t-18\de}}\le \rho^{18\de}\qquad\mbox{and}\\ \nonumber && \Big(1- \frac{1}{\w_7} \Big) \frac{30\de A+ B}{B}+ \frac{64D\de L \big(\sqrt{\la}\big)^{t-32\de}}{B\rho^{t-36\de}}\le \rho^{36\de} \end{eqnarray} are satisfied for all $1\le l\le 30\de$ and all $t\ge 0$. The above inequalities rewrite as \begin{eqnarray} \label{rho1} && \big( A(l-1)+ B\big)\,\rho^{t-18\de}+ 32D\de L\left(\sqrt{\la}\right)^{t-32\de} \le \big(Al+ B\big)\,\rho^t \qquad \mbox{and}\\ \label{rho2} && \Big(1- \frac{1}{\w_7} \Big) (30\de A+ B)\,\rho^{t-36\de}+ 64D\de L\left(\sqrt{\la}\right)^{t-32\de}\le B\,\rho^t \end{eqnarray} and they are satisfied for all $1\le l\le 30\de$ and all $t\ge 0$. The proof of the proposition proceeds by induction on $d(a,b)+d(a,b')$. We consider the following two cases. \noi {\bf \underline{Case~1}.} {\it \ $(a|b)_{b'}> 10\de$\quad or\quad $(a|b')_b > 10\de$.} \setlength{\unitlength}{1 cm} \begin{figure}[h] \begin{center} \begin{picture}(8,4.5) %\graphpaper[1](0,0)(8,4.5) \put(0.5,1.5){\circle*{0.1}} \put(0.15,1.6){\small$a$} \put(0.5,1.2){\circle*{0.1}} \put(0.05,1){\small$a'$} \put(7,4){\circle*{0.1}} \put(7.2,4){\small$b$} \put(6,0){\circle*{0.1}} \put(6.2,-.1){\small$b'$} %from a to b \qbezier(0.5,1.5)(5,2)(7,4) \put(2.6,2.3){\footnotesize$p[b,a]$} %from a' to b \qbezier(0.5,1.2)(5.4,1.9)(7,4) \put(4.3,1.8){\footnotesize$p[b,a']$} %from a to b' \qbezier(0.5,1.5)(4.5,2)(6,0) %from a' to b' \qbezier(0.5,1.2)(4.5,1.9)(6,0) %inscribed triangle \put(4.55,2.45){\circle*{0.1}} \put(4.55,1.145){\circle*{0.1}} \put(6.1,2.8){\tiny$10\de$} %v \put(5,2.64){\circle*{0.1}} %v' \put(5.1,2.49){\circle*{0.1}} \put(5.7,3.9){\circle*{0.1}} \put(5.3,3.9){\small$x$} %from v to x \qbezier(5,2.64)(5.35,3.27)(5.7,3.9) \end{picture} \end{center} \caption{\label{case1-AB-rho} Proposition~\ref{AB-rho}, Case~1.} \end{figure} Without loss of generality, $(a|b')_b> 10\de$ (interchange $b$ and $b'$ otherwise). The 0-cycle $f(b,a)- f(b,a')$ can be uniquely represented as $f_+- f_-$, where $f_+$ and $f_-$ are 0-chains with non-negative coefficients, disjoint supports, and of the same $\ell^1$-norm. We have $$f(b,a)=f_0+ f_+ \quad {\rm and} \quad f(b,a')=f_0+ f_-$$ for some 0-chain $f_0$ with non-negative coefficients (actually $f_0= \min \big\{f(b,a),f(b,a')\big\}$). Denote $\al:= |f_+|_1= |f_-|_1= \ep(f_+)= \ep(f_-)$, where $\ep$ is the augmentation map. Since $d(a,a')\le 1$, then \begin{eqnarray*} (a|a')_b &&\ge \frac{1}{2}\, \big[ d(a,b)+d(a',b)- 1\big]\\ &&\ge \frac{1}{2}\, \big[ d(a,b)+d(a,b')-32\de\big], \end{eqnarray*} and by Proposition~\ref{c-c-c}(5), \begin{eqnarray} \label{alpha} \al&&= \frac{1}{2}\,\Big| f(b,a)- f(b,a')\Big|_1\\ &&\le \frac{1}{2}L\la^{(a|a')_b}\nonumber \\ &&\le \frac{1}{2}L\left(\sqrt{\la}\right)^{d(a,b)+d(a,b')-32\de}. \nonumber \end{eqnarray} By the definition of hyperbolicity in section~\ref{s_hyp} and the assumption $(a|b')_b>10\de$, we have $$ d\big(p[b,a](10\de), p[b,a'](10\de)\big) \leq \de.$$ Hence there exists a vertex $x_0\in B\big(p[b,a](10\de),8\de\big)\cap B\big(p[b,a'](10\de),8\de\big)$. By the definitions of $r$ and~$\fff$, \begin{eqnarray*} && \big| r(a,b)- r(a',b)- r(a,b')+ r(a',b') \big|\\ && = \Big| r\big(a,\fff(b,a)\big)+ 1- r\big(a',\fff(b,a')\big)- 1- r(a,b')+ r(a',b') \Big|\\ && = \Big| r\big(a,star(f_0+ f_+)\big) - r\big(a',star(f_0+ f_-)\big) - r(a,b')+ r(a',b') \Big|\\ && \le \Big| r\big(a,star(f_0)+\al x_0\big)- r\big(a',star(f_0)+ \al x_0\big) - r(a,b')+ r(a',b') \Big| +\\ && + \Big| r\big(a, star(f_+)- \al x_0\big) \Big| + \Big| r\big(a',\al x_0- star(f_-)\big) \Big|.\\ \end{eqnarray*} \noi Now we bound each of the three terms in the last sum. We number these terms consecutively as $T_1, T_2, T_3$. \noi {\it \underline{Term~$T_1$}.} \ Using the same argument as in Case 1 in the proof of Proposition 5, one checks that, for each $$x\in supp\big( star(f_0)+ \al x_0 \big) \se B(p[b,a](10\de),8\de)\cap B(p[b,a'](10\de),8\de),$$ the following conditions hold: \begin{eqnarray*} && d(x,b')\le d(b,b')-1\le 30\de \qquad {\rm and} \\ && d(a,b)+ d(a,b')- 18\de \le d(a,x)+d(a,b') \le d(a,b)+d(a,b')-1. \end{eqnarray*} In particular, the induction hypotheses are satisfied for the vertices $a,a',x,b'$, giving \begin{eqnarray*} && \big| r(a,x)- r(a',x)- r(a,b')+ r(a',b') \big| \\ && \le \big( A\ d(x,b')+ B\big)\,\rho^{d(a,x)+d(a,b')}\\ &&\le \Big(A\big(d(b,b')-1\big)+ B\Big)\,\rho^{d(a,b)+d(a,b')-18\de}. \end{eqnarray*} Since $star(f_0)+\al x_0$ is a convex combination, by linearity of $r$ in the second variable, \begin{eqnarray*} T_1=&& \Big| r\big(a,star (f_0)+\al x_0\big)- r\big(a',star(f_0)+ \al x_0\big)- r(a,b')+ r(a',b') \Big|\\ \le && \Big(A\, \big(d(b,b')-1\big)+ B\Big)\,\rho^{d(a,b)+d(a,b')-18\de}. \end{eqnarray*} \noi {\it \underline{Terms~$T_2$ and~$T_3$}.} \ Since $star(f_+)- \al x_0$ is a 0-cycle supported in a ball of radius~$8\de$, by Proposition~\ref{D} and inequality~(\ref{alpha}), \begin{eqnarray*} T_2= && \Big| r\big(a, star(f_+)- \al x_0\big) \Big|\\ \le && D\ \Big| star(f_+)- \al x_0 \Big|_1\cdot 16\de \\ \le && D\cdot 2\al\cdot 16\de\\ \le && 16D\de L\left(\sqrt{\la}\right)^{d(a,b)+d(a,b')-32\de}. \end{eqnarray*} Analogously, \begin{equation*} T_3= \Big| r\big(a',\al x_0- star(f_-)\big) \Big|\le 16D\de L\left(\sqrt{\la}\right)^{d(a,b)+d(a,b')-32\de}. \end{equation*} Combining the three bounds above and using the definition of $B$ and $\rho$ (inequality~(\ref{rho1})), \begin{eqnarray*} && \big| r(a,b)- r(a',b)- r(a,b')+ r(a',b') \big|\\ &&\le T_1+ T_2+ T_3 \\ && \le \Big( A\,\big(d(b,b')-1\big)+B \Big)\,\rho^{d(a,b)+d(a,b')-18\de}+ 32D\de L\left(\sqrt{\la}\right)^{d(a,b)+d(a,b')-32\de}\\ && \le \big( A\,d(b,b')+ B\big)\,\rho^{d(a,b)+d(a,b')}. \end{eqnarray*} This finishes Case~1. \noi{\bf \underline{Case~2}.} {\it \ $(a|b)_{b'}\le 10\de$\quad and\quad $(a|b')_b\le 10\de$.} \setlength{\unitlength}{1 cm} \begin{figure}[h] \begin{center} \begin{picture}(9,4) %\graphpaper[1](0,0)(9,4) \put(0.5,2){\circle*{0.1}} \put(0,1.95){\small$a$} \put(0.5,1.7){\circle*{0.1}} \put(0,1.6){\small$a'$} \put(8,3.5){\circle*{0.1}} \put(8.2,3.5){\small$b$} \put(7,0.5){\circle*{0.1}} \put(7.2,0.2){\small$b'$} %from a to b \qbezier(0.5,2)(6,2)(8,3.5) %from a to b' \qbezier(0.5,2)(6,2)(7,0.5) %from a' to b \qbezier(0.5,1.7)(6,1.8)(8,3.5) %from a' to b' \qbezier(0.5,1.7)(6,1.8)(7,0.5) %inscribed triangle \put(5.93,2.57){\circle*{0.1}} \put(5.9,1.31){\circle*{0.1}} \put(6.5,3.2){\scriptsize$10\de\!\ge$} \put(6.6,1.1){\scriptsize$\le\!\! 10\de$} %w \put(4.4,2.25){\circle*{0.1}} %v \put(5.2,2.39){\circle*{0.1}} \put(4.95,2.5){\small$v$} \put(5.25,2.23){\circle*{0.1}} %v' \put(4.4,1.72){\circle*{0.1}} \put(4,1.68){\small$v'$} \put(4.38,1.53){\circle*{0.1}} \put(5.6,3.7){\circle*{0.1}} \put(5.2,3.6){\small$x$} \put(5,0.5){\circle*{0.1}} \put(5.2,0.3){\small$x'$} %from v to x \qbezier(5.2,2.39)(5.4,3.045)(5.6,3.7) %from v' to x' \qbezier(4.4,1.72)(4.7,1.11)(5,0.5) %from v' to w \qbezier(4.4,1.72)(4.4,2)(4.4,2.25) \put(4.45,1.86){\scriptsize $\le\!\! \de$} \end{picture} \end{center} \caption{\label{case2-AB-rho} Proposition~\ref{AB-rho}, Case~2.} \end{figure} As in Case~1, we have \begin{eqnarray} \nonumber & f(b,a)- f(b,a')= f_+- f_-,\qquad f(b,a)=f_0+ f_+,\quad f(b,a')= f_0+ f_-, &\\ \nonumber & \al:= |f_+|_1= |f_-|_1=\ep(f_+)= \ep(f_-), &\\ \nonumber & \ds\al\le \frac{1}{2}L\la^{(a|a')_b}\le \frac{1}{2}L\left(\sqrt{\la}\right)^{d(a,b)+d(a,b')-32\de},& \end{eqnarray} where $f_+$, $f_-$, and $f_0$ are 0-chains with non-negative coefficients, and $f_+$ and $f_-$ have disjoint supports. Analogously, interchanging $b$ and $b'$, \begin{eqnarray*} & f(b',a)- f(b',a')= f'_+- f'_-,\qquad f(b',a)=f'_0+ f'_+,\quad f(b',a')=f'_0+ f'_-, &\\ & \al':= |f'_+|_1= |f'_-|_1=\ep(f'_+)= \ep(f'_-), &\\ & \ds\al'\le \frac{1}{2}L\la^{(a|a')_{b}}\le \frac{1}{2}L\left(\sqrt{\la}\right)^{d(a,b)+d(a,b')-32\de},& \end{eqnarray*} where $f'_+$, $f'_-$, and $f'_0$ are 0-chains with non-negative coefficients, and $f'_+$ and $f'_-$ have disjoint supports. Denote $v:=p[b,a](10\de)$ and $v':=p[b',a](10\de)$. By the conditions of Case~2 and $\de$-hyperbolicity of $\G$, using the same argument as in Case 2 in the proof of Proposition~\ref{N}, we obtain $d(v, v') \leq 11\de$. Let $x_0$ be a vertex closest to the mid-point of a geodesic path connecting $v$ to $v'$. Proposition~\ref{c-c-c}(2) implies that \begin{eqnarray*} \label{x0} && supp\ f_0\cup supp\ f'_0\se B(x_0, 7\de)\qquad\mbox{and}\\ && supp\ f_-\cup supp\ f_+\cup supp\ f'_-\cup supp\ f'_+ \se B(x_0, 8\de). \end{eqnarray*} By the definition of $r$, \begin{eqnarray*} && \big| r(a,b)- r(a',b)- r(a,b')+ r(a',b') \big|\\ && = \Big| r\big(a,\fff(b,a)\big)- r\big(a',\fff(b,a')\big)- r\big(a,\fff(b',a)\big)+ r\big(a',\fff(b',a')\big) \Big|\\ && \le \Big| r\big(a,star(f_0+ f_+)\big)- r\big(a',star(f_0+ f_-)\big)- r\big(a,star(f'_0+ f'_+)\big)+ r\big(a',star(f'_0+ f'_-)\big) \Big|\\ && \le \Big| r\big(a, star(f_0)+ \al x_0 - star(f'_0)- \al' x_0 \big)- r\big(a', star(f_0)+ \al x_0 - star(f'_0)- \al' x_0 \big)\Big| +\\ && + \Big| r\big(a, star(f_+)\big)- r\big(a,\al x_0\big) \Big| + \Big| r\big(a',\al x_0\big)- r\big(a',star(f_-)\big) \Big|+\\ &&+ \Big| r\big(a,\al' x_0\big)- r\big(a, star(f'_+)\big) \Big| + \Big| r\big(a',star(f_-')\big)- r\big(a',\al' x_0\big) \Big|.\\ \end{eqnarray*} \noi Now we bound each of the five terms in the last sum. We number these terms consecutively as $S_1$, ..., $S_5$. \noi {\it \underline{Term~$S_1$}.} One checks that, for each \begin{eqnarray*} && x\in supp\big( star(f_0)+ \al x_0 \big) \se B(v,8\de)\cap B(p[b,a'](10\de),8\de)\quad \mbox{and}\\ && x'\in supp\big( star(f'_0)+ \al' x_0 \big) \se B(v',8\de)\cap B(p[b',a'](10\de),8\de), \end{eqnarray*} the following conditions hold: \begin{eqnarray*} && d(x,x')\le 30\de \quad \mbox{and}\\ && d(a,b)+ d(a,b')- 36\de \le d(a,x)+d(a,x') \le d(a,b)+d(a,b')-1. \end{eqnarray*} In particular, the induction hypotheses are satisfied for the vertices $a,a',x,x'$, giving \begin{eqnarray} \label{ind2} && \big| r(a,x)- r(a',x)- r(a,x')+ r(a',x') \big| \\ \nonumber && \le \Big( A\ d(x,x')+ B\Big)\,\rho^{d(a,x)+d(a,x')}\\ \nonumber && \le \Big(30\de A+ B\Big)\,\rho^{d(a,b)+d(a,b')-36\de}. \end{eqnarray} Recall that $\w_7$ is the number of vertices in a ball of radius $7\de$. Let $\be$ be the (positive) coefficient of $x_0$ in the 0-chain $star(f_0)$, and $\be'$ be the (positive) coefficient of $x_0$ in the 0-chain $star(f'_0)$. Without loss of generality, we can assume $\big|star(f_0)\big|_1\le \big|star(f'_0)\big|_1$. Since $x_0$ was chosen so that $supp\ f_0\cup supp\ f'_0\se B(x_0,7\de)$, by the definition of $star$, we have \begin{eqnarray*} &&\be= {\frac{1}{\w_7}}\, \big|f_0\big|_1= {\frac{1}{\w_7}}\, \big|star(f_0)\big|_1 \le {\frac{1}{\w_7}}\, \big|star(f'_0)\big|_1= {\frac{1}{\w_7}}\, \big|f'_0\big|_1= \be'\qquad\mbox{and}\\ &&\al -\al'= \Big( 1- \big|f_0 \big|_1\Big)- \Big( 1- \big|f'_0 \big|_1\Big)= \big|f'_0 \big|_1- \big|f_0 \big|_1= \w_7(\be'-\be)\ge 0. \end{eqnarray*} Therefore, \begin{eqnarray*} && \Big| star(f_0)+ \al x_0- star(f'_0)- \al' x_0 \Big|_1\\ && \le \Big| star(f_0)- \be x_0 \Big|_1 + \Big| \be' x_0- star(f'_0) \Big|_1 + \Big| \big[ (\al- \al')- (\be' -\be) \big] x_0 \Big|_1\\ && = \Big( \big| star(f_0) \big|_1 - \be \Big)+ \Big( \big| star(f'_0) \big|_1 - \be' \Big)+ \big( \be' -\be \big) \big( \w_7-1 \big)\\ && = \Big( \big| f_0 \big|_1 - \be \Big)+ \Big( \big| f'_0 \big|_1 - \be' \Big)+ \Big( \big|f'_0 \big|_1- \big|f_0 \big|_1 \Big) \Big(1- \frac{1}{\w_7} \Big)\\ && = \big| f_0 \big|_1 \Big(1- \frac{1}{\w_7} \Big) + \big| f'_0 \big|_1 \Big(1- \frac{1}{\w_7} \Big) + \Big( \big|f'_0 \big|_1- \big|f_0 \big|_1 \Big) \Big(1- \frac{1}{\w_7} \Big)\\ && = 2 \big| f'_0 \big|_1 \Big(1- \frac{1}{\w_7} \Big)\\ && \le 2 \Big(1- \frac{1}{\w_7} \Big). \end{eqnarray*} Since $\big[ star(f_0)+ \al x_0 \big]- \big[ star(f'_0)+ \al' x_0 \big]$ is a 0-cycle, it is of the form $h_+- h_-$, where $h_+$ and $h_-$ are 0-chains with non-negative coefficients, disjoint supports and of the same $\ell^1$-norm, so we can define $$\g := |h_+|_1= |h_-|_1= \ep(h_+)= \ep(h_-).$$ By the above inequality, $$\g= {\frac{1}{2}}\ |h_+- h_-|_1\le 1- \frac{1}{\w_7},$$ then, by~(\ref{ind2}) and linearity of $r$ in the second variable, \begin{eqnarray*} S_1\ = && \Big| r(a, h_+- h_-)- r(a',h_+- h_-) \Big|\\ = && \Big| r(a,h_+)- r(a',h_+)- r(a,h_-)+ r(a',h_-) \Big|\\ \le && \g\cdot \Big(30\de A+ B\Big)\,\rho^{d(a,b)+d(a,b')-36\de}\\ \le && \Big(1- \frac{1}{\w_7} \Big) \Big(30\de A+ B\Big)\,\rho^{d(a,b)+d(a,b')-36\de}. \end{eqnarray*} \noi {\it \underline{Terms~$S_2-S_5$}.} \ Analogously to term~$T_2$ in Case~1, \begin{eqnarray*} & S_2\ \le 16D\de L\left(\sqrt{\la}\right)^{d(a,b)+d(a,b')-32\de},\qquad S_3\ \le 16D\de L\left(\sqrt{\la}\right)^{d(a,b)+d(a,b')-32\de},&\\ & S_4\ \le 16D\de L\left(\sqrt{\la}\right)^{d(a,b)+d(a,b')-32\de},\qquad S_5\ \le 16D\de L\left(\sqrt{\la}\right)^{d(a,b)+d(a,b')-32\de}.&\\ \end{eqnarray*} Combining the bounds for the five terms above and using the definition of $B$ and $\rho$ (inequality~(\ref{rho2})), \begin{eqnarray*} && \big| r(a,b)- r(a',b)- r(a,b')+ r(a',b') \big|\\ && \le S_1+ S_2+ S_3+ S_4+ S_5\\ && \le \Big(1- \frac{1}{\w_7} \Big) (30\de A+ B)\,\rho^{d(a,b)+d(a,b')-36\de}+ 64D\de L\left(\sqrt{\la}\right)^{d(a,b)+d(a,b')-32\de}\\ && \le B\,\rho^{d(a,b)+d(a,b')}\\ &&\le \big( A\,d(b,b')+ B\big)\, \rho^{d(a,b)+d(a,b')}. \end{eqnarray*} Proposition~\ref{AB-rho} and Theorem~\ref{r} are proved. \qed %\input{bch-g.txt} \section{More properties of $r$.} In this section, we prove two distance-like inequalities for the function $r$ introduced in the previous section. As before, let $G$ be a hyperbolic group and $\Gamma$ be the Cayley graph of $G$ with respect to a finite generating set. For any subset $A\subseteq \Gamma$, denote $$N_G(A,R) :=\{ x\in G\ \big|\ d(x, A)\leq R\}.$$ \begin{ppp} \label{C1} There exists $C_1\geq 0$ with the following property. If $a,b\in G$, $\g$ is a geodesic in~$\G$ connecting $a$ and $b$, $x\in G\cap\g$, $\g'$ is the part of $\g$ between $x$ and $b$, and $c\in N_G (\g', 9\delta)$, then $$\big|r(a,c)-r(a,x)-r(x,c)\big| \leq C_1 \qquad (Figure~\ref{fig-axc}).$$ \end{ppp} \setlength{\unitlength}{.7cm} \begin{figure}[h] \begin{center} \begin{picture}(12,4.2) %\graphpaper[1](0,0)(12,4.2) \put(1,1.5){\circle*{0.15}} \put(.3,1.4){\small$a$} \put(11,1.5){\circle*{0.15}} \put(11.4,1.35){\small$b$} \put(8,2.5){\circle*{0.15}} \put(8.4,2.35){\small$c$} \put(8,1.5){\circle*{0.15}} \put(8.3,1){\small$c'$} \put(4,1.5){\circle*{0.15}} \put(4.3,1){\small$x$} \put(6,0.5){\footnotesize$\g$} %from a to b \qbezier(1,1.5)(6,1.5)(11,1.5) %from x to c \qbezier(4,1.5)(6,2)(8,2.5) %from a to c \qbezier(1,1.5)(4.5,1.7)(8,2.5) \end{picture} \end{center} \caption{\label{fig-axc} Proposition~\ref{C1}.} \end{figure} \pf Let $$C_1 :=(80\de+N+36\de DL)\sum_{k=0}^{\infty} \lambda^{k-18\delta},$$ where $L\geq 1$ and $0< \lambda <1$ are as in Propositions~\ref{c-c-c}(5) and~\ref{la'}(5), $N$ is as in Proposition~\ref{N}, and $D$ is as in Proposition~\ref{D}. It suffices to show the inequality $$\big|r(a,c)- r(a,x)-r(x,c)\big| \leq (80\de+N+36\de DL)\sum_{k=0}^{d(x,c)} \lambda^{k-18\delta}.$$ We will prove it by induction on $d(x,c)$. If $d(x,c)\leq 40\delta$, by Proposition 4 and Theorem 6(2) we have \begin{eqnarray*} &&\big|r(a,c)-r(a,x)-r(x,c)\big| \leq \big|r(a,c)-r(a,x)\big| +r(x,c)\\ && \leq \big(d(c,x)+N \big)+ d(x,c) \leq 80\de+ N\\ && \leq (80\de+ N+36\de DL) \sum_{k=0}^{d(x,c)} \la^{k-18\de}. \end{eqnarray*} Now we assume that $d(x,c)> 40 \delta.$ There exists a vertex $c'\in \g'$ with $d(c',c)\leq 9\delta$, so $$d(a,c)\geq d(a,c')-9\delta \geq d(x,c')-9\delta \geq d(x,c)-18\delta >10\delta.$$ Hence by the definition of the function $r$, we have $$r(a,c)=r\big(a,\bar{f}(c,a)\big)+1\qquad\mbox{and}\qquad r(x,c)=r\big(x,\bar{f}(c,x)\big)+1.$$ Also \begin{eqnarray*} (a|x)_c &&= \frac{1}{2}\big[d(c,a)+d(c,x)-d(a,x)\big]\\ &&\ge\frac{1}{2} \big[d(c',a)- 9\de+ d(c',x)- 9\delta- d(a,x)\big]\\ &&=d(x,c')-9\delta \\ &&\geq d(x,c)-18\delta. \end{eqnarray*} By Proposition~\ref{la'}(5), $$\Big|\bar{f}(c,x)-\bar{f}(c,a)\Big|_1\leq L \lambda^{(a|x)_c}\leq L\lambda^{d(x,c)-18\delta}.$$ This, together with Proposition~\ref{D} and Proposition 2(2), implies that \begin{eqnarray*} \Big| r\big(a,\bar{f}(c,a)\big)- r\big(a,\bar{f}(c,x)\big)\Big| &&= \Big| r\big(a,\bar{f}(c,a)- \bar{f}(c,x)\big)\Big|\\ &&\leq DL \lambda^{d(x,c)-18\delta} \, diam\big(supp (\fff(c,a)-\fff(c,x))\big)\\ && \le 36\de DL \la^{d(x,c)-18\de}. \end{eqnarray*} By Proposition~\ref{la'}(2) and~\ref{la'}(7), for every $y\in supp(\bar{f}(c,x))$, we have $$ d(x,y)\leq d(x,c)-1\qquad\mbox{and}\qquad y\in N_G (\g', 9\delta).$$ Hence by the induction hypotheses, we obtain \begin{eqnarray*} &&\big| r(a,c)-r(a,x)-r(x,c)\big|\\ &&= \Big| \big(r\big(a,\bar{f}(c,a)\big)+1\big)-r(a,x)-\big(r(x,\bar{f}(c,x))+1\big)\Big|\\ &&\leq \Big|r\big(a,\bar{f}(c,a)\big)-r\big(a, \bar{f}(c,x)\big)\Big| +\Big|r\big(a,\bar{f}(c,x)\big)- r(a,x)- r\big(x,\bar{f}(c,x)\big)\Big|\\ &&\leq 36\de DL\la^{d(x,c)-18\de}+ (80\de+ N+ 36\de DL) \sum_{k=0}^{d(x,c)-1}\lambda^{k-18\delta}\\ &&\le (80\de+ N+ 36\de DL) \sum_{k=0}^{d(x,c)}\lambda^{k-18\delta}. \end{eqnarray*} \qed \begin{ppp} \label{M'} There exists $M'\geq 0$ such that $$\big|r(a,b)-r(a',b)\big| \leq M'\,d(a,a')$$ for all $a, a', b \in G$. \end{ppp} \pf Recall that $\delta\geq 1$. Let $$M':=(20\delta+ 3+ 36\de DL) \sum_{k=0}^{\infty} \lambda^{k-19\de}.$$ The Cayley graph $\Gamma$ is a geodesic metric space, hence it suffices to show the inequality $\big|r(a,b)-r(a',b)\big|\le M'$ in the case when $d(a,a')=1$. For that, it suffices to show the inequality $$\big|r(a,b)-r(a',b)\big| \leq (20\delta+3+36\de DL)\sum_{k=0}^{d(b,a)}\lambda^{k-19\de} $$ when $d(a,a')=1$. We will prove it by induction on $d(a,b)$. If $d(a,b)\leq 10\delta +1$, then by Theorem 6(2) we have \begin{eqnarray*} \big|r(a,b)-r(a',b)\big| && \leq r(a,b)+ r(a',b)\\ && \leq d(a,b)+ d(a',b)\\ && \leq 20\delta +3 \\ &&\le (20\delta+ 3+ 36\de DL)\sum_{k=0}^{d(b,a)} \la^{k-19\de}. \end{eqnarray*} If $d(a,b)>10\delta +1$, then $d(a',b)>10\delta.$ For every $y\in supp(\fff(b,a))\cup supp(\fff(b,a'))$, by Proposition 2(2) we have $$(a|a')_y=\frac{1}{2}\big[d(y,a)+d(y,a')-d(a,a')\big]\ge d(b,a)-19\de.$$ Hence by the definition of the function $r$, the induction hypothesis and Propositions~\ref{la'}(5) and ~\ref{D}, we obtain \begin{eqnarray*} && \big|r(a,b)-r(a',b)\big|\\ &&= \Big|\big(r(a, \bar{f}(b,a))+1\big)- \big(r(a', \bar{f}(b,a'))+1\big)\Big|\\ &&\leq \big|r(a,\bar{f}(b, a))-r(a', \bar{f}(b,a))\big|+\big|r(a',\bar{f}(b,a))-r(a',\bar{f}(b,a'))\big|\\ &&\leq (20\delta+ 3+ 36\de DL)\sum_{k=0}^{d(b,a)-1} \la^{k-19\de}+ DL\la^{d(b,a)-19\de} diam\Big(supp\big (\fff(b,a)-\fff(b,a')\big)\Big)\\ &&\leq (20\delta+ 3+ 36\de DL)\sum_{k=0}^{d(b,a)-1} \la^{k-19\de}+ 36\de DL\la^{d(b,a)-19\de}\\ &&\le (20\delta+ 3+ 36\de DL)\sum_{k=0}^{d(b,a)} \la^{k-19\de}. \end{eqnarray*} \qed \section{Definition and properties of a new metric $\hat{d}$.} In this section, we use the function $r$ defined in section 3 to construct a $G$-invariant metric $\hat{d}$ on a hyperbolic group $G$ such that $\hat{d}$ is quasi-isometric to the word metric and prove that $(G, \hat{d})$ is weakly geodesic and strongly bolic. We define $$s(a,b):=\frac{1}{2}\, \big[r(a,b)+ r(b,a)\big]$$ for all $ a,b \in G$. \begin{ppp} \label{uvw} The above function $s$ satisfies the following conditions. \begin{itemize} \item [(a)] There exists $M\ge 0$ such that $$\big|s(u,v)-s(u,v')\big|\le M\, d(v,v')\quad\mbox{and}\quad \big|s(u,v)-s(u',v)\big|\le M\, d(u,u')$$ for all $u, u', v,v'\in G$. \item [(b)] There exists $C_1\ge 0$ such that if a vertex $w$ lies on a geodesic connecting vertices~$u$ and~$v$, then $$\big| s(u,v)-s(u,w)-s(w,v)\big|\le C_1.$$ \end{itemize} \end{ppp} \pf {\bf (a)} Since $s$ is symmetric, it suffices to show only the first inequality. Since the Cayley graph $\Gamma$ is a geodesic metric space, it suffices to consider only the case $d(v,v')=1$. This case follows from Propositions~\ref{N} and~\ref{M'}. {\bf (b)} follows from Proposition~\ref{C1}. \qed \begin{ppp} \label{C2} There exists $C_2\geq 0$ such that $$s(a,b)\leq s(a,c)+ s(c,b)+C_2$$ for all $ a, b, c\in G$. \end{ppp} \pf Let $\bar{a} \in p[b,c], \bar{c} \in p[a,b], \bar{b}\in p[a,c]$ such that $$d(b, \bar{c})= d(b, \bar{a}), \quad d(c, \bar{a})= d(c, \bar{b}), \quad d(a, \bar{c})= d(a, \bar{b}).$$ By the definition of hyperbolicity, we have $$ d(\bar{a}, \bar{b})\leq \delta, \quad d(\bar{a}, \bar{c})\leq \delta, \quad d(\bar{b}, \bar{c})\leq \delta.$$ By Proposition~\ref{uvw}, \begin{eqnarray*} s(a,b) && \leq s(a,\bar{c}) + s(\bar{c}, b) +C_1\\ &&\leq \big(s(a,\bar{b})+ M\,d(\bar{b},\bar{c})\big)+ \big( s(\bar{a},b) + M\,d(\bar{c},\bar{a}) \big)+ C_1\\ &&\le s(a,\bar{b})+s(\bar{a},b)+ 2\de M + C_1\\ &&\le \big(s(a,\bar{b})+ s(\bar{b},c)\big) +\big(s(c,\bar{a})+ s(\bar{a},b)\big)+ 2\de M+ C_1\\ &&\le s(a,c)+ s(c,b)+ 2\de M+ 3C_1, \end{eqnarray*} so we set $C_2:= 2\de M+ 3C_1$. \qed For every pair of elements $a, b\in G$, we define \[ \hat{d}(a,b):=\left\{ \begin{array}{ll} s(a, b) +C_2 & \mbox{if $a\neq b$,} \\ 0 & \mbox{if $a=b$.} \end{array} \right. \] \begin{ppp} The function $\hat{d}$ defined above is a metric on $G$. \end{ppp} \pf By definition, $\hat{d}$ is symmetric, and $\hat{d}(a,b)=0$ iff $a=b$. The triangle inequality is a direct consequence of Proposition~\ref{C2}. \qed \begin{ppp} \label{d0} There exist constants $C\ge 0$ and $0\le\mu<1$ with the following property. For all $R\ge 0$ and all \mbox{$a, a', b, b'\in G$} with $d(a,a')\le R$ and $d(b,b')\le R$, $$\big| \hat{d}(a,b)- \hat{d}(a',b) - \hat{d}(a,b')+ \hat{d}(a',b') \big|\le R^2 C\mu^{d(a,b)-2R}.$$ In particular, if $d(a,a')\le R$ and $d(b,b')\le R$, then $$\hat{d}(a,b)-\hat{d}(a', b)- \hat{d}(a,b')+\hat{d}(a',b') \rightarrow 0 \quad\mbox{as}\quad d(a,b)\rightarrow \infty.$$ \end{ppp} \pf Take $C$ and $\mu$ as in Theorem~\ref{r}(3). Increasing $C$ if needed we can assume that $a\neq b, a\neq b', a'\neq b, a'\neq b'$. If $ a=a'$ or $b=b'$, then $$\hat{d}(a,b)-\hat{d}(a', b)- \hat{d}(a,b')+\hat{d}(a',b')=0.$$ If $d(a,a')=1$ and $d(b,b')=1$, then by Theorem~\ref{r}(3), \begin{eqnarray*} &&\big|\hat{d}(a,b)-\hat{d}(a', b)- \hat{d}(a,b')+\hat{d}(a',b')\big|\\ &&= \big|s(a,b)-s(a',b)-s(a,b')+s(a',b')\big|\\ &&\le C\mu^{d(a,b)}. \end{eqnarray*} Without loss of generality, we can assume that $R$ is an integer. In the general case $$d(a,a')\le R\qquad \mbox{and}\qquad d(b,b')\le R,$$ pick vertices $a=a_0, a_1, ... , a_R=a'$ with $d(a_{i-1}, a_i)\le 1$ and $b=b_0, b_1, ... , b_R=b'$ with $d(b_{j-1},b_j)\le 1$ and note that $d(a_i,b_j)\ge d(a,b)-2R$. Then we have \begin{eqnarray*} &&\big|\hat{d}(a,b)-\hat{d}(a', b)- \hat{d}(a,b')+\hat{d}(a',b')\big|\\ &&= \big|s(a,b)-s(a',b)-s(a,b')+s(a',b')\big| \\ &&= \Big|\sum_{i=1}^{R} \sum_{j=1}^{R} \big( s(a_{i-1},b_{j-1})-s(a_i,b_{j-1})-s(a_{i-1},b_j)+s(a_i,b_j)\big)\Big| \\ &&\le \sum_{i=1}^{R} \sum_{j=1}^{R} \big|s(a_{i-1},b_{j-1})-s(a_i,b_{j-1})-s(a_{i-1},b_j)+s(a_i,b_j)\big| \\ &&\le R^2 C\mu^{d(a,b)-2R}. \end{eqnarray*} \qed Recall that a metric space $(X, d)$ is said to be {\sf weakly geodesic}~\cite{KS, KS1} if there exists $\de_1\geq 0$ such that, for every pair of points $x$ and $y$ in $X$ and every $t\in [0, d(x,y)]$, there exists a point $a\in X$ such that $d(a,x)\leq t+\de_1$ and $d(a,y)\leq d(x,y)-t+\de_1$. \begin{ppp} \label{weakly-geodesic} The metric space $(G, \hat{d})$ is weakly geodesic. \end{ppp} \pf Let $x, y\in G$ and $z\in G\cap p[x,y]$. By the definition of $\hat{d}$ and Proposition~\ref{uvw}(b), we have $$ \hat{d}(x,z) +\hat{d}(z,y)-\hat{d}(x,y)\leq C_1+2C_2.$$ It follows that $$ \hat{d}(x,z)\leq \hat{d}(x,y)+ C_1 + 2C_2,$$ hence the image of the map $$\hat{d}(x, \cdot): G\cap p[x,y] \rightarrow [0, \infty),$$ is contained in $[0,\hat{d}(x,y) +C_1 +2C_2].$ Also, the image contains 0 and $\hat{d}(x,y)$. By Proposition~\ref{uvw}(a), we have $$\big|\hat{d}(x, z')-\hat{d}(x, z)\big|\leq M$$ when $d(z', z)=1$. This, together with the fact that $p[x,y]$ is a geodesic path, implies that the image of the map $$\hat{d}(x, \cdot): G\cap p[x,y] \rightarrow [0, \hat{d}(x,y)+C_1+2C_2]$$ is $M$-dense in $[0, \hat{d}(x,y)]$, i.e. for every $t\in [0, \hat{d}(x,y)]$, there exists $a \in G\cap p[x,y]$ such that $$\big|\hat{d}(x,a)-t\big| \leq M.$$ It follows that $\hat{d}(x,a)\leq t+ M$, and by Proposition~\ref{uvw}(b) we also have $$\big| \hat{d}(x,y)- \hat{d}(x,a)-\hat{d}(a,y)\big| \leq C_1 +2C_2.$$ This implies that \begin{eqnarray*} \hat{d}(a,y) && \leq \hat{d}(x,y)-\hat{d}(x,a) +C_1 +2C_2\\ &&\leq \hat{d}(x,y) -t +M+C_1+2C_2. \end{eqnarray*} Therefore $(G,\hat{d})$ is weakly geodesic for $\de_1:=M+C_1+2C_2$. \qed Kasparov and Skandalis introduced the concept of bolicity in~\cite{KS, KS1}. \begin{ddd} A metric space $(X,d)$ is said to be {\sf bolic} if there exists $\de_2\ge 0$ with the following properties: \begin{itemize} \item [(B1)] for any $R>0$, there exists $R'>0$ such that for all $a, a',b,b' \in X$ satisfying $$d(a,a')+d(b,b')\leq R\qquad\mbox{and}\qquad d(a,b)+d(a',b')\geq R',$$ we have $$d(a,b')+d(a',b)\leq d(a, b)+ d(a', b') +2\de_2;\qquad\mbox{and}$$ \item [(B2)] there exists a map $m: X\times X \rightarrow X$, such that, for all $x,y, z\in X$, we have $$2d(m(x,y), z)\leq \big(2d(x,z)^2 +2d(y,z)^2-d(x,y)^2\big) ^{\frac{1}{2}} +4\de_2.$$ \end{itemize} $(X,d)$ is called {\sf strongly bolic} if it is bolic and the above condition (B1) holds for every $\de_2> 0$~\cite{L3}. \end{ddd} \begin{ppp} \label{bolic} The metric space $(G, \hat{d})$ is strongly bolic. \end{ppp} \pf Proposition~\ref{d0} yields condition~(B1) for all $\de_2>0$. It remains to show that there exist $\de_2\ge 0$ and a map $m: G\times G \rightarrow G$, such that, for all $x,y, z\in G$, we have $$2\hat{d}(m(x,y), z)\leq \big(2\hat{d}(x,z)^2 +2 \hat{d}(y,z)^2-\hat{d}(x,y)^2\big)^{\frac{1}{2}} +4\de_2.$$ By Proposition~\ref{weakly-geodesic} and its proof, there exists a vertex $m(x,y)\in G\cap p[x,y]$ such that \begin{equation} \label{mxy} \Big|\hat{d} (x, m(x,y))-\frac{\hat{d}(x,y)}{2}\Big|\leq \delta_1\quad\mbox{and} \quad\Big|\hat{d}(m(x,y),y)-\frac{\hat{d}(x,y)}{2} \Big| \leq \delta_1. \end{equation} By the definition of $\delta$-hyperbolicity, we know that either \begin{itemize} \item [(1)] there exists $a\in G\cap p[z,y]$ such that $d(m(x,y), a)\leq \delta +1$, or \item [(2)] there exists $b\in G \cap p[x,z]$ such that $d(m(x,y), b)\leq \delta +1.$ \end{itemize} In case (1), we have $$\big|\hat{d}(z, m(x,y))-\hat{d}(z,a)\big| \leq \hat{d}(m(x,y), a)\leq \delta +1+C_1,$$ $$\big|\hat{d}(y, m(x,y))-\hat{d}(y,a)\big| \leq \hat{d}(m(x,y),a)\leq \delta +1+C_1. $$ Hence, by Proposition~\ref{uvw}(b), we obtain \begin{eqnarray*} \hat{d}(z, m(x,y))+\hat{d}(x,y) &&\leq \hat{d}(z,a)+\delta +1+C_1 +\hat{d}(x,y)\\ &&\le \hat{d}(z,a)+ \delta +1+C_1 +\hat{d}(x, m(x,y))+ \hat{d}(m(x,y),y)\\ &&\leq \hat{d}(z,a)+\hat{d}(a,y)+\hat{d}(x, m(x,y))+ 2\delta +2C_1+2\\ &&\leq \hat{d}(y,z)+\hat{d}(x, m(x,y))+\delta', \end{eqnarray*} where $\delta':=2\delta +3C_1 +2$. In case (2), we similarly have $$ \hat{d}(z, m(x,y))+ \hat{d}(x,y) \leq \hat{d}(x,z)+\hat{d}(m(x,y),y) +\delta'.$$ It follows from~(\ref{mxy}) that \begin{eqnarray*} \hat{d}(z, m(x,y))+\hat{d}(x,y) && \leq \sup\big\{\hat{d}(x,z)+\hat{d}(y, m(x,y)),\, \hat{d}(y,z) +\hat{d}(x, m(x,y))\big\} +\delta'\\ &&\leq \sup \big\{\hat{d} (x,z), \hat{d}(y,z)\big\} +\frac{\hat{d}(x,y)}{2} +\delta_1 +\delta'. \end{eqnarray*} Hence \begin{equation} \label{2d} 2\hat{d}(z, m(x,y)) \leq 2 \sup \big\{\hat{d}(x,z), \hat{d}(y,z)\big\}- \hat{d}(x,y)+ 4\delta_2, \end{equation} where $\delta_2 :=\frac{\delta_1 +\delta'}{2}.$ If $t, u,$ and $v$ are non-negative real numbers such that $|u-v|\leq t$, then $$(2u-v)^2 \leq 2u^2+2t^2 -v^2.$$ Setting $t:=\inf \{\hat{d}(x,z), \hat{d}(y,z)\}, u:=\sup\{\hat{d}(x,z), \hat{d}(y,z)\}, v:=\hat{d}(x,y),$ we obtain $$2 \sup\big\{\hat{d}(x,z), \hat{d}(y,z)\big\}-\hat{d}(x,y) \leq \big(2\hat{d}(x,z)^2+2 \hat{d}(y,z)^2 -\hat{d}(x,y)^2\big)^{\frac{1}{2}}.$$ Therefore, by~(\ref{2d}), $$2\hat{d}(z, m(x,y))\leq \big(2\hat{d}(x,z)^2 +2 \hat{d}(y,z)^2-\hat{d}(x,y)^2\big)^{\frac{1}{2}} +4\delta_2.$$ \qed We summarize the results of this section. \begin{ttt} \label{metric} Every hyperbolic group $G$ admits a metric $\hat{d}$ with the following properties. \begin{itemize} \item [(1)] $\hat{d}$ is $G$-invariant, i.e. $\hat{d}(g\cdot x,g\cdot y)=\hat{d}(x,y)$ for all $x,y,g\in G$. \item [(2)] $\hat{d}$ is quasiisometric to the word metric $d$, i.e. there exist $A>0$ and $B\geq 0$ such that $$\frac{1}{A} \hat{d}(x,y)-B \leq d(x,y) \leq A \hat{d}(x,y) +B$$ for all $x,y\in G$. \item [(3)] The metric space $(G,\hat{d})$ is weakly geodesic and strongly bolic. \end{itemize} \end{ttt} \section{The Baum-Connes conjecture for hyperbolic groups.} In this section, we combine Theorem 17 with Lafforgue's work to prove the main result of this paper. \begin{ddd} An action of a topological group $G$ on a topological space $X$ is called {\sf proper} if the map $G\times X\to X\times X$ given by $(g,x)\mapsto (x,gx)$ is a proper map, that is the preimages of compact subsets are compact. \end{ddd} When $G$ is discrete, an action is proper iff it is properly discontinuous, i.e. if the set $\{g\in G\ \big|\ K\cap gK \not=\emptyset\}$ is finite for any compact $K\se X$. The following deep theorem was proved by Lafforgue using Banach KK-theory. \begin{ttt}[Lafforgue~\cite{L3}] If a discrete group $G$ has property RD, and G acts properly and isometrically on a strongly bolic, weakly geodesic, and uniformly locally finite metric space, then the Baum-Connes conjecture holds for $G$. \end{ttt} \begin{ttt} \label{bc} The Baum-Connes conjecture holds for hyperbolic groups and their subgroups. \end{ttt} \pf Let $H$ be a subgroup of a hyperbolic group $G$. By Theorem~\ref{metric}(2), there exist constants $A> 0$ and $B\ge 0$ such that $d(a,b)\leq A\,\hat{d}(a,b)+B$ for all $ a, b\in G$. Hence $(G, \hat{d})$ is uniformly locally finite and the $H$-action on $(G,\hat{d})$ is proper. By Theorem~\ref{metric}, $(G, \hat{d})$ is weakly geodesic and strongly bolic, and the $H$-action on $(G,\hat{d})$ is isometric. By a theorem of P. de la Harpe and P. Jolissaint, $H$ has property RD~\cite{h,jp}. Now Theorem 19 implies Theorem 20. \qed Theorem 20 has been proved independently by Vincent Lafforgue using a different and elegant method~\cite{L4}. The following result is a direct consequence of Theorem~\ref{bc}. \begin{ttt} The Kadison-Kaplansky conjecture holds for any torsion free subgroup $G$ of a hyperbolic group, i.e. there exists no non-trivial projection in the reduced group $C^{\ast}$-algebra $C_r^{\ast}(G)$. \end{ttt} Recall that an element $p$ in $C_r^{\ast}(G)$ is said to be a projection if $p^{\ast}=p, p^2 =p$. A projection in $C^{\ast}_r(G)$ is said to be non-trivial if $p\neq 0, 1$. It is well known that the Baum-Connes conjecture for a torsion free discrete group $G$ implies the Kadison-Kaplansky conjecture for $G$~\cite{BC, BCH}. Michael Puschnigg has independently proved Theorem 21 using a beautiful local cyclic homology method~\cite{p}. 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