%% Gersten: ag(odd).tex %% Allcock: ag(even).tex %% Each change of text file, raise indicator by 2 \input amstex \documentstyle{amsppt} %\input epsf.tex %% This removes the AMS logo from bottom of first page \catcode`\@=11 \def\logo@{} \catcode`\@=\active \pageheight{46pc} \NoBlackBoxes \tolerance10000 \mag\magstep1 \def\Z{\Bbb Z} \def\R{\Bbb R} \def\Q{\Bbb Q} \def\wt{\widetilde} \def\H#1{H_{(#1)}} \def\sk#1{^{(#1)}} \def\cd{\hbox{\rm cd}} \def\Area{\hbox{\rm Area}} \def\normal{\vartriangleleft} \def\bs{\bigskip} \def\ss{\smallskip} %% Format for including .epsf files \iffalse $$ \hbox{% \epsfxsize=0.7\hsize \epsfbox{char-fig1.eps} } $$ \fi %% This removes the AMS logo from bottom of first page \catcode`\@=11 \def\logo@{} \catcode`\@=\active %\syntax %\hoffset.2in \def\epi{\twoheadrightarrow} \def\mono{\rightarrowtail} \def\implies{\Rightarrow} \def\Def{\definition{Definition}} \def\endDef{\enddefinition} \def\Area{\hbox{\rm Area}} \def\Sl{\hbox{\rm Sl}}\def\Gl{\hbox{\rm Gl}} \def\Diam{\hbox{\rm Diam}} \mag\magstep 1 \NoBlackBoxes\tolerance10000 \def\ms{\medskip} \def\ss{\smallskip} \def\bs{\bigskip} \def\<{\langle} \def\>{\rangle} \def\one{^{(1)}} \def\two{^{(2)}} \def\zero{^{(0)}} \def\sk#1{^{(#1)}} \def\inv{^{-1}} \def\n{^{(n)}} \def\normal{\vartriangleleft} \def\nnormal{\triangleleft} \def\wt{\widetilde} \def\wh{\widehat} \TagsAsMath \def\ni{\noindent} \def\A{\Bbb A} \def\Z{\Bbb Z} \def\R{\Bbb R} \def\N{\Bbb N} \def\Q{\Bbb Q} \def\a{\alpha} \def\aa{\alpha} \def\bb{\beta} \def\d{\delta} \def\ee{\epsilon} \def\e{\epsilon} \def\g{\gamma} \def\G{\Gamma} \def\x{\xi} \def\z{\zeta} \def\p{\pi} \def\l{\lambda} \def\o{\omega} \def\k{\kappa} \def\m{\mu} \def\n{\nu} \def\s{\sigma} \def\t{\tau} \def\H{\Bbb H} \def\C{\Bbb C} \def\AA{\Cal A} \def\BB{\Cal B} \def\CC{\Cal C} \def\DD{\Cal D} \def\EE{\Cal E} \def\HH{\Cal H} \def\JJ{\Cal J} \def\KK{\Cal K} \def\OO{\Cal O} \def\SS{\Cal S} \def\TT{\Cal T} \def\UU{\Cal U} \def\VV{\Cal V} \def\WW{\Cal W} \def\XX{\Cal X} \def\YY{\Cal Y} \def\ZZ{\Cal Z} \def\PP{\Cal P} \def\QQ{\Cal Q} \def\LL{\Cal L} \def\MM{\Cal M} \def\NN{\Cal N} \def\RR{\Cal R} \def\DD{\Cal D} \def\FF{\Cal F} \def\GG{\Cal G} \def\Fix{\hbox{\rm Fix}} \def\Out{\hbox{\rm Out}} \def\Aut{\hbox{\rm Aut}} \def\SO{\hbox{\rm SO}} \def\U{\hbox{\rm U}} %\input option_keys \def\G{\Gamma} \def\D{\Delta} \def\L{\Lambda} \def\P{\Pi} \def\th{^{\hbox{\rm th }}} \def\st{^{\hbox{\rm st }}} \def\rd{^{\hbox{\rm rd }}} \def\supo{\mathaccent"7017 } %% This is how to make the small "o" that goes over a set %% to indicate its interior. \def\degrees{^\circ} %% This is how I want to make eg 60 degrees in $ $. \def\eg{{{\it e.g.},\/}} \def\Eg{{{\it E.g}.,\/}} \def\ie{{{\it i.e.}\/}} \def\Ie{{{\it I.e.}\/}} \def\etal{{\it et al.\/}} \def\etc{{\it etc.\/}} \def\via{{\it via\/}} \def\cf{{\it cf.\ }} \def\apriori{{\it a priori\/}} \def\afortiori{{\it a fortiori\/}} \def\qv{{{\it q.v.},\/}} \def\vide{{{\it vide},\/}} \def\h1{H^{(1)}} \def\c1{C^{(1)}} \def\z1{Z^{(1)}} \def\b1{B^{(1)}} %\syntax \def\hinf{H_{(\infty)}} \def\zinf{Z_{(\infty)}} \def\cinf{C_{(\infty)}} \def\binf{B_{(\infty)}} \def\G{\Gamma} \def\Area{\hbox{\rm Area}} \def\con{\hbox{\rm Cone$_\omega$}} \def\res{\hbox{\rm res}} \def\warea{\hbox{\rm w-Area}} \topmatter \title {A homological characterization of hyperbolic groups}\endtitle \author{D. J. Allcock and S. M. Gersten} \endauthor \leftheadtext{\null} \rightheadtext{\null} %\keywords %\endkeywords %\subjclass %\endsubjclass %\date %\enddate \abstract\nofrills{{\bf Abstract.\/} A finitely presented group~$G$ is hyperbolic iff $\h1_1(G,\R)=0=\bar\h1_2(G,\R)$, where $\h1_*$ (resp. $\bar\h1_*$) denotes the $\ell_1$-homology (resp. reduced $\ell_1$-homology). If $\G$ is a graph, then every $\ell_1$ 1-cycle in $\G$ with real coefficients can be approximated by 1-cycles of compact support. A 1-relator group~$G$ is hyperbolic iff $\h1_1(G,\R)=0$.} \endabstract \address{Mathematics Department, University of Utah, Salt Lake City, UT 84112, USA} \endaddress \email{allcock\@math.utah.edu, gersten\@math.utah.edu} \endemail %\email{gersten\@math.utah.edu} %\endemail \thanks{The first author is supported by an NSF postdoctoral fellowship and the second author is partially supported by NSF grant DMS-9500769} \endthanks \keywords{hyperbolic group, linear isoperimetric inequality, $\ell_1$-homology}\endkeywords \subjclass{20F05, 20F32, 57M07}\endsubjclass \footnote"\null"{\copyright D. J. Allcock and S. M. Gersten 1997, all rights reserved} \endtopmatter \document \subhead{\S 1. Introduction} \endsubhead %\baselineskip=24pt In \cite{Ge1} the second author showed that a finitely presented group~$G$ is word hyperbolic iff a certain cohomology group vanishes, namely $\hinf^2(G,\ell_\infty)=0$. G.~A.~Swarup asked whether there are analogous vanishing theorems in homology. Our main result, Corollary~4.8, is that a finitely presented group~$G$ is word hyperbolic (henceforth called hyperbolic) iff $\h1_1(G,\R)=0=\bar\h1_2(G,\R)$, where $\h1_*$ denotes the (unreduced) $\ell_1$-homology and where $\bar\h1_*$ denotes the reduced $\ell_1$-homology; their definitions are recalled in \S 2. It is not clear whether there is any relation between vanishing theorems for $\ell_\infty$-cohomology and those for $\ell_1$-homology, for while it is true that $\ell_\infty$ is the dual of $\ell_1$, the Banach space $\ell_1$ is not reflexive. For example, $\hinf^1(G,\R)$ and the reduced group $\bar\hinf^1(G,\R)$ are both nonzero for every infinite finitely generated group~$G$. Our main technical tool, Theorem~3.3, states that every $\ell_1$ 1-cycle with real coefficients on a graph can be approximated by 1-cycles of compact support. We give in 6.1 below an example due to E. Formanek, which shows that there is no analog of Theorem~3.3 in general for higher dimensional cycles on a complex. In \S 5 we show that if $G$ is either a 1-relator group or the fundamental group of a finite piecewise Euclidean 2-complex of nonpositive curvature, then $\h1_2(G,\R)=0$. It follows from this and from Theorem~4.5 that such a group $G$ is hyperbolic iff $\h1_1(G,\R)=0$. We are grateful to P. Brinkmann and K. Whyte for helpful comments. \subhead{\S 2. The $\ell_1$-homology and $\ell_\infty$-cohomology of a group} \endsubhead A norm on an abelian group $A$ is a function $|\ \ |:A\to \R$ satisfying $|-a|=|a|$, $|a+a'|\le |a|+|a'|$, and $|a|\ge 0$ with $|a|=0$ iff $a=0$, for all $a,a'\in A$. We recall that a group~$G$ is said to be of type~$\Cal F_n$ if there is a CW-complex~$X'$ of type~$K(G,1)$ with finite $n$-skeleton. For example, $G$ is of type~$\Cal F_1$ iff it is finitely generated and of type~$\Cal F_2$ iff it is finitely presented. If $G$ is a group of type $\Cal F_{n+1}$, let $X'$ be a CW-complex of type $K(G,1)$ with finite $(n+1)$-skeleton and let $X$ be the universal cover of $X'$. A summable $i$-chain~$f$ on $X$ with values in $A$ is a skew-symmetric function from the oriented $i$-cells of $X$ with values in $A$ (so $f(\bar e)=-f(e)$ where $e$ is an $i$-cell and $\bar e$ is the same geometric $i$-cell with the opposite orientation \footnote {Precisely, $\bar e$ is obtained from $e$ by precomposing the characteristic mapping of $e$ with some fixed orientation reversing involution of the $i$-cell.}) such that $\sum_e |f(e)|<\infty,$ where the sum is over all oriented $i$-cells~$e$. It is convenient to think of the chain~$f$ as an infinite sum $\sum_{e\in \Cal O} f(e)e$ where $\Cal O$ is an orientation on the $i$-cells, that is $\Cal O$ contains precisely one of the pair $e$, $\bar e$ for each oriented $i$-cell~$e$. Then we define the $\ell_1$-norm~$|f|$ of $f$ by $$ |f|=\sum_{e\in\Cal O} |f(e)|.\tag 2.1 $$ If $i\le n+1$ then because $X'$ has only finitely many $i$-cells, there is an upper bound on the $\ell_1$-norms of the boundaries $\partial e$ of $i$-cells $e$. This means that if $\c1_i(X,A)$ denotes the set of summable $i$-chains with values in $A$, then the function~$\partial$ extends to a continuous homomorphism $\hat\partial:\c1_i(X,A)\to \c1_{i-1}(X,A)$. One checks that $\hat\partial^2=0$, so we have a chain complex defined in a range of degrees $i\le n+1$. The $\ell_1$-homology $\h1_i(X,A)$ is defined in the usual way as $\z1_i(X,A)/\b1_i(X,A)$, where $\z1_i(X,A)$ is the subgroup of summable $i$-cycles and where $\b1_i$ is the image of $\hat\partial:\c1_{i+1}(X,A)\to \c1_i(X,A)$. This makes sense for $i\le n$. %\footnote{ %Note in this connection that if $X$ is an arbitrary simplicial %complex, then $\h1_i(X,A)$ is defined for all $i\ge 0$. %} Now we consider the $\ell_\infty$ cohomology groups. We define $\cinf^i( X,A)$ to be the subgroup of cellular $i$-cochains~$h$ such that there is a number $M_h>0$ with $|h(\sigma)|\le M_h$ for all $i$-cells $\sigma$. It follows from the finiteness of ${X'}^{(n+1)}$ that the coboundary $\delta h$ lies in $\cinf^{i+1}( X,A)$ if $i\leq n$. One defines cocycles $\zinf^i$, coboundaries $\binf^i=\delta(\cinf^{i-1})$, and cohomology groups $\hinf^i=\zinf^i/\binf^i$ in the usual way. This definition makes sense for all $i\leq n+1$, since one does not require the finiteness conditions on the $(n+2)$-cells to formulate the condition $\delta h=0$ for $h\in\cinf^{n+1}(X,A)$. The value of the $\ell_1$-homology and $\ell_\infty$-cohomology groups arises from their quasi-isometry invariance. \footnote{For the notion of quasi-isometry of metric spaces and of finitely generated groups see \cite{GH}.} It is known that the condition that a group~$G$ have type $\Cal F_n$ is a quasi-isometry invariant \cite{Al}\cite{Gr2}. It is also known that if $X'$ and $Y'$ are a $K(G,1)$ and a $K(G',1)$, respectively, both with finite $(n+1)$-skeleta, and if the groups $G$ and $G'$ are quasi-isometric, then there are isomorphisms $\h1_i(X,A)\cong\h1_i(Y,A)$ for $i\leq n$ and $\hinf^i(X,A)\cong\hinf^i(Y,A)$ for $i\leq n+1$; here $X$ and $Y$ are the universal covers of $X'$ and $Y'$ respectively. A proof of these facts can be constructed along the lines of \cite{Ge2} \S 11. We may thus unambiguously define $\h1_i(G,A)$ as $\h1_i(X,A)$ and $\hinf^i(G,A)$ as $\hinf^i(X,A)$, and we note that the vanishing of either of these groups is a {\it geometric property\/} in the sense that it is an invariant of quasi-isometry type. In this paper we shall be interested mainly in the case $A=\R$. For $i\le n+1$, $\z1_i(X,\R)$ is a closed subspace of $\c1_i(X,\R)$ since it is the kernel of the bounded linear operator $\hat\partial:\c1_i(X,\R)\to \c1_{i-1}(X,\R)$. However, the image $\b1_i(X,\R)$ need not be a closed subspace. It is usual to define the reduced $\ell_1$-homology $\bar\h1_i(X,\R)$ to be $\z1_i(X,\R)/\bar B_i(X,\R)$, where $\bar B_i(X,\R)$ is the closure of $B_i(X,\R)$ in the normed linear space $\c1_i(X,\R)$ under the $\ell_1$-norm, defined in (2.1) above. $\bar\h1_i(X,\R)$ is defined for $i\le n+1$ since the closure operation $\bar B_{n+1}(X,\R)$ does not depend on the continuity of $\partial_{n+2}$, and it is quasi-isometry invariant in the range where it is defined. % We can now state the main result of this paper. % \proclaim{Theorem 2.2} If $G$ is a hyperbolic group, the % $\h1_1(G,\R)=0$ and $\bar \h1_1(G,\R)=0$. % \endproclaim % The proof will be given in \S 4 below. \subhead{\S 3. The approximation theorem for summable $1$-cycles} \endsubhead We will use Serre's formulation of a graph $\G$ as consisting of two sets, its vertices $V(\G)$ and edges $E(\G)$. The set $E(\G)$ is equipped with a fixed-point-free involution $e\mapsto\bar e$ as well as the initial-vertex map $\iota:E(\G)\to V(\G)$. One defines the terminal vertex $\tau(e)$ of an edge to be $\iota(\bar e)$. Serre's notion of an edge corresponds to what is usually called a directed edge, since one of its vertices is distinguished as its initial vertex and the other as its terminal vertex. A path is a sequence $(e_i)$ of edges satisfying $\tau e_i=\iota e_{i+i}$ for all $i$. A path is simple if the vertices $\iota e_i$ are all distinct. A circuit is a path $(e_1,\ldots,e_n)$ with $\tau e_n=\iota e_1$, up to the equivalence relation that two paths represent the same circuit if one is obtained by the other by cyclic permutation of its edges. Since our notion of edge includes a direction, these concepts might also be called ``directed paths'' and ``directed circuits''. %An orientation %of $\G$ is a subset of $E(\G)$ meeting $\{e,\bar e\}$ %exactly once, for each edge $e$. A directed graph is a graph %together with an orientation of it; %we conform to the usual practice of calling a {\it directed %edge\/} one which %lies in the orientation. This is the usual convention, for putting %an arrow on a geometric edge of a 1-dimensional CW-complex %has the effect of choosing one of two possible orientations %(\ie equivalence classes of parametrizations) on that geometric %edge. A real-valued $1$-cochain is a function $f:E(\G)\to\R$ such that $f(e)=-f(\bar e)$ (skew-symmetry) for all $e\in E(\G)$. For each vertex $v$ of $\G$ we define the divergence of $f$ at $v$ to be $$ \text{Div}_v(f)=\sum_{\iota e=v} f(e), $$ provided that the sum converges absolutely. The 1-cochain~$f$ is called a 1-chain if $f$ has finite support. %, which is the same as saying that the support of the %geometric realization of $f$ is compact. %If $e_1,\ldots,e_n$ are edges of $\G$, then the chain denoted by %$\sum_{i=1}^ne_i$ is defined to be the function on $E(\G)$ whose %value on an edge $e$ equals %$$ %\text{the cardinality of $\{i:e_i=e\}$}- %\text{the cardinality of $\{i:e_i=\bar e\}$}. %$$ If $e$ is an edge of $\G$ then $e$ is identified with the 1-chain which takes the value~1 on~$e$, $-1$ on $\bar e$, and is zero on all other edges of~$\G$. A $1$-cochain $f$ is called, by a slight abuse of terminology, a summable (or $\ell_1$) 1-chain if it is summable as a function on $E(\G)$. (This definition of a summable chain agrees with that given in \S2.) In this case, $\text{Div}_v(f)$ is defined for all vertices $v$, and we observe that $f$ is a summable 1-cycle iff $\text{Div}_v(f)=0$ for all vertices~$v$. We will write $\G_+(f)$ for the set of edges on which $f$ takes positive values. Here is a geometric interpretation of these concepts. A cochain may be regarded as a ``flow'' along the edges of $\G_+(f)$, with the magnitude of the flow along an edge $e$ given by $f(e)$. An edge of $\G_+(f)$ is an edge of $\G$ which points in the direction of the flow. $\text{Div}_v(f)$ is the net flow out of the vertex $v$, and $f$ is a cycle just if the substance flowing is conserved. The following lemma is a combinatorial analog of Stokes' theorem. \proclaim{Lemma 3.1} If $\G$ is a graph and $f$ is a summable $1$-chain on $\G$, then $$ \sum_v\hbox {\rm Div}_v(f)=0. $$ \endproclaim \demo{Proof} For each $e\in E(\G)$ the term~$f(e)$ occurs in the sum $\text{Div}_{\iota e}(f)$ and the term $f(\bar e)=-f(e)$ occurs in the sum $\text{Div}_{\tau e}(f)$. By absolute convergence we may rearrange terms freely; after doing so, all terms cancel. \enddemo \proclaim{Lemma 3.2} If $\G$ is a graph and $f$ is a summable 1-cycle on $\G$ such that $\G_+(f)$ contains no nontrivial circuits, then $f=0$. \endproclaim \demo{Proof} Suppose that $f(e_0)=A>0$ for some edge $e_0$. Let $\G'$ be the subgraph of $\G$ defined by the condition that $e,\bar e\in\G'$ if either $e$ or $\bar e$ lies in a path of $\G_+(f)$ that begins at $\iota e_0$; that is, $\G'$ is the smallest subgraph of $\G$ that contains all edges of all paths in $\G_+(f)$ which begin at $e_0$. Let $f'$ be the summable $1$-chain on $\G$ which coincides with $f$ on $\G'$ and vanishes elsewhere. If $v$ is a vertex of $\G'$ then an edge of $\G_+(f)$ whose initial (resp. terminal) vertex is $v$ does (resp. might) lie in $\G_+(f')$. This shows that $\text{Div}_v(f')\ge \text{Div}_v(f)=0$ for all vertices $v$. Lemma 3.1 shows that $\sum_{v}\text{Div}_v(f')=0$, so we conclude that $\text{Div}_v(f')=0$ for all $v$. Taking $v=\iota e_0$ we see that $\iota e_0$ must be the terminal vertex of some edge of $\G_+(f')$ and hence of $\G_+(f)$. This contradicts the hypothesis that $\G_+(f)$ contains no circuits, completing the proof. \enddemo We turn now to the main result of this section. We call a family~$\Cal F$ of 1-cocycles {\it coherent\/} if for any two 1-cocycles $f$ and $g$ in $\Cal F$ and for each edge~$e$ we have $f(e)g(e)\ge 0$. That is, if both $f(e)$ and $g(e)$ are nonzero then they have the same sign. If $c=(e_1,\ldots,e_n)$ is a simple circuit in $\G$ then we construct the $1$-cycle $\sum_{i=1}^ne_i$, which we will also call $c$ and refer to as a simple circuit. This duplicate definition of $c$ should cause no confusion, since the (simple) circuit may be reconstructed from the $1$-cycle. \proclaim{Theorem 3.3} If $\G$ is a graph and $f$ is a summable real valued 1-cycle on $\G$, then $f$ can be approximated arbitrarily closely in the $\ell_1$-sense by finite sums of simple circuits with real coefficients. More precisely, there is a countable coherent family $C$ of simple circuits and a function $g:C\to[0,\infty)$ so that $f=\sum_c g(c)c$, where the convergence of the sum is monotone. \endproclaim \demo{Proof} The support of $f$ is countable, since $f$ is summable; this shows that the family of simple directed circuits in $\G_+(f)$ is countable. We take $C$ to be this family, which is obviously coherent. The space of functions from $C$ to $[0,\infty)$ is partially ordered in the obvious way: $g\geq h$ if $g(c)\geq h(c)$ for all $c\in C$. To each summable $g:C\to[0,\infty)$ there is associated the cocycle $z_g$ (not necessarily summable) on $\G$ defined by $$ z_g=\sum_{c\in C}g(c)c. $$ The convergence is monotone because $g(c)\geq0$ for all $c$ and $C$ is a coherent family. Consider the set $\Cal S$ of functions $g:C\to[0,\infty)$ satisfying $$ 0\leq z_g(e)\leq f(e)\text{ for all $e\in\G_+(f)$}. \tag 3.3.1 $$ A standard Zorn's lemma argument shows that $\Cal S$ has a maximal element, say $g$. We will complete the proof by showing $z_g=f$. By (3.3.1), $\G_+(f-z_g)$ is a subset of $\G_+(f)$. If $\G_+(f-z_g)$ contained a simple circuit, say $c_0$, then letting $a$ be the smallest value taken by $f-z_g$ on any edge of $c_0$ we would find that the function $g':C\to[0,\infty)$ defined by $$ g'(c)= \left\{\eqalign{\rlap{$g(c)+a$,}\hskip1in&\hbox{if $c=c_0$;}\cr \rlap{$g(c)$}\hskip1in&\hbox{otherwise,}\cr}\right. $$ lies in $\Cal S$ and strictly dominates $g$, contradicting the maximality of $g$. Therefore $\G_+(f-z_g)$ contains no simple circuits and thus no circuits. By lemma 3.2 we have $f-z_g=0$, completing the proof. \enddemo \demo{Remark} It is natural to ask whether theorem~3.3 admits generalizations, allowing approximations of summable $n$-chains in CW complexes. Example~6.1 below, due to E. Formanek, shows that this is not possible in general. \enddemo \proclaim{Corollary 3.4} If $T$ is a tree, then every summable 1-cycle on $T$ is zero. \endproclaim If $\G$ is an arbitrary graph then $Z_1(\G,\R)$ is a closed subspace of $C_1(\G,\R)$ for the $\ell_1$-norm topology, so we can consider its completion $\hat Z_1(\G,\R)\subset \c1(\G,\R)$. The continuity of the boundary map $\partial:C_1(\G,\R)\to C_0(\G,\R)$ shows that $\hat Z_1(\G,\R)\subseteq \z1(\G,\R)$. We have \proclaim{Corollary 3.5} For every graph~$\G$, $\hat Z_1(\G,\R)=\z1(\G,\R)$. \endproclaim \demo{Proof} If $f\in\z1(\G,\R)$ then it follows from Theorem~3.3 that $f$ can be approximated by 1-cycles of compact support. Since $\hat Z_1(\G,\R)$ is closed in $\c1(\G,\R)$, it follows that $f\in \hat Z_1(\G,\R)$. \enddemo \subhead{\S 4. Hyperbolic groups} \endsubhead A hyperbolic group~$G$ is a finitely presented group satisfying the linear isoperimetric inequality for fillings of edge-circuits in its Cayley graph \cite{GH}\cite{Ge1}. If $X'$ is a space of type $K(G,1)$ with finite 2-skeleton and $X$ is the universal cover of~$X'$, then $G$ is hyperbolic iff there exists~$K>0$ (the isoperimetric constant) so that for all $z\in Z_1(X,\Z)$ there exists $c\in C_2(X,\Z)$ with $\partial c=z$ and $|c|\le K|z|$ \cite{Ge5}. The norms here are both $\ell_1$-norms, for a basis of oriented $i$-cells for $C_i(X,\Z)$, for $i=1,2$. For the remainder of this section we abbreviate $H_i(X,\R)$, $\h1_i(X,\R)$, $B_i(X,\R)$, etc., by $H_i$, $\h1_i$, $B_i$, etc. \proclaim{Proposition 4.1} If $G$ is a hyperbolic group, then $\h1_1=0$. \endproclaim \demo{Proof} Given $f\in\z1_1$ we must show that $f=\partial c$ for some $c\in \c1_2 $. By Theorem~3.3 we can write $f=\sum_{i\ge 1}a_iz_i$ where $\{z_i\mid i\ge 1\}$ is a coherent family of compactly supported integral 1-cycles on $X$ and where $a_i\ge 0$ for all $i$. By hyperbolicity there are $c_i\in C_2(X,\Z)$ with $\partial c_i=z_i$ and $|c_i|\le K|z_i|$ for all $i\ge 1$, where $K$ is the isoperimetric constant. Let $c=\sum_ia_ic_i$. We calculate $|c|\le \sum_i a_i|c_i|\le K\sum_ia_i|z_i|=K|\sum_ia_iz_i| =K|f|$, where the second-to-last equality follows from the coherence of the $z_i$. In particular this shows that $c$ is a summable 2-cochain and justifies the interchange of summation and boundary map in the calculation $\partial c=\sum_i a_i\partial c_i=\sum_ia_iz_i=f$. This completes the proof that $\h1_1=0$. \enddemo The boundary map $\partial$ provides a surjection $j:C_2\to B_1$, which induces a norm, called the filling norm, on $B_1$. The completion of $B_1$ with respect to this norm is denoted $\hat B_1$, and $j$ extends to a continuous map (in fact a surjection; see below) $\hat\jmath:\c1_2\to\hat B_1$. The identity map $i:B_1\to Z_1$ is continuous, where $B_1$ (resp. $Z_1$) is equipped with the filling (resp. $\ell_1$) norm, and therefore extends to a continuous map $\hat\imath:\hat B_1\to\z1_1$. \proclaim{Lemma 4.2} The sequence $$ 0\to\z1_2/\hat Z_2\to\hat B_1 @>\hat\imath>> \z1_1\to \h1_1\to0 \tag 4.2.1 $$ of linear maps of vector spaces is exact. {\rm(}The second map is induced by the restriction to $\z1_2$ of the completion $\hat\jmath$ of the bounday map $j:C_2\to B_1$.{\rm)} \endproclaim \demo{Proof} We will use several times the fact that if $C$ is a normed vector space and $Z$ is a closed subspace, then the completion of $C/Z$ coincides with the quotient of the completion of $C$ by the closure therein of $Z$, \ie\ $\widehat{C/Z}=\hat C/\hat Z $, where $\hat{\phantom{x}}$'s denote completions. The last paragraph of the proof of theorem~1.5.3 in \cite{KR} is essentially a proof of this assertion. Since $B_1=C_1/Z_1$, passing to completions shows that the natural map $\c1_2/\hat Z_2\to \hat B_1$ is an isomorphism. Therefore the restriction of this map to $\z1_2/\hat Z_2$ is injective, proving exactness at the term $\z1_2/\hat Z_2$ of 4.2.1. The map $\partial:C_2\to C_1$ factors as the composition $i\circ j:C_2 @>j>> B_1 @>i>> Z_1\subset C_1$. Therefore the completion $\hat\partial$ factors as $\hat\imath\circ\hat\jmath$. Since $\hat\jmath$ is surjective (by the fact above), we have the exact sequence $$ 0\to\text{Ker}(\hat\jmath) \to\text{Ker}(\hat\partial) \to\text{Ker}(\hat\imath)\to0. $$ Since $\text{Ker}(\hat\jmath)$ is the kernel of the completion $\hat\jmath:\c1_2\to\hat B_1$, namely $\hat Z_2$, and $\text{Ker}(\hat\partial)=\z1_1$ by definition, we have $\text{Ker}(\hat\imath)=\z1_2/\hat Z_2$, proving exactness of 4.2.1 at $\hat B_1$. Because $\hat\partial=\hat\imath\circ\hat\jmath$ with $\hat\jmath$ surjective, we see that $\text{Im}(\hat\partial)=\text{Im}(\hat\imath)$. Therefore $\z1_1/\text{Im}(\hat\imath)=\z1_1/\b1_1=\h1_1$. This proves exactness at the terms $\z1_1$ and $\h1_1$ of 4.2.1, completing the proof. \enddemo \proclaim{Lemma 4.3} If $G$ is of type $\Cal F_3$, then there is an exact sequence $$ 0\to \hat Z_2/\b1_2\to \h1_2\to \z1_2/\hat Z_2\to 0,\tag 4.3.1 $$ where $\hat Z_2$ denotes the closure of $Z_2$ in $\c1_2$. \endproclaim \demo{Proof} This follows from the filtration $\b1_2\subseteq\hat Z_2\subseteq\z1_2.$ \enddemo \demo{Remark} The term $\z1_2/\hat Z_2$ in (4.3.1) represents the obstruction to approximating a summable 2-cycle by 2-cycles of compact support. The term $\hat Z_2/\b1_2$, as we shall see shortly, represents the obstruction to the linear isoperimetric inequality holding for 2-cycles. \enddemo \proclaim{Proposition 4.4} If $G$ is a hyperbolic group then $\h1_2=0$. \endproclaim \demo{Proof} By a result of Rips \cite{GH} hyperbolic groups are of type $\Cal F_\infty$, so we can take $X'$ a $K(G,1)$ with finite $n$-skeleton for all~$n$ and let $X=\wt {X'}$ as usual. Now $B_1=Z_1$ as normed linear spaces over $\R$, where $B_1$ is equipped with the filling norm, so $\hat B_1=\hat Z_1=\z1_1$, where the last equality follows from Corollary~3.5. Thus the map~$\hat\imath$ in (4.1.1) is an isomorphism, and it follows that $\z1_2/\hat Z_2=0$. Thus every summable 2-cycle can be approximated by 2-cycles of compact support. By \cite{AB} a hyperbolic group satisfies a linear isoperimetric inequality for fillings of compactly supported 2-cycles. By approximation it follows that the linear isoperimetric inequality holds for fillings of summable 2-cycles. It follows that the term $\hat Z_2/\b1_2=0$. Thus $\h1_2=0$ by Lemma~4.3, and the proof is complete. \enddemo We can now state our main result \proclaim{Theorem 4.5} A finitely presented group~$G$ is hyperbolic iff $\h1_1= 0$ and $\z1_2/\hat Z_2=0$. \endproclaim \demo{Remark} The assertion that $\z1_2=\hat Z_2$ is that every summable 2-cycle on $X$ can be approximated by real 2-cycles of compact support. If $G$ is of type~$\Cal F_2$ then $\bar\h1_2(G,\R)$ is defined (see \S 2) even though $\h1_2$ may not be defined, and it is equal to $\z1_2/\bar Z_2$, where $\bar Z_2=\bar B_2=\hat Z_2$ is the closure of $Z_2$ in $\c1_2$. If $G$ is of type~$\Cal F_3$, then $\h1_2$ is defined, and one has the exact sequence (4.3.1) exhibiting $\bar\h1_2=\z1_2/\bar Z_2$ as a quotient of $\h1_2$. \enddemo \demo{Proof of Theorem~4.5} The vanishing of $\h1_1$ and $\z1_2/\hat Z_2$ for a hyperbolic group follow from the preceding lemmas. We assume now that $G$ is of type~$\Cal F_2$ with $\h1_1= 0$. The argument that follows is a reduction to the main result of \cite{Ge1}, that a finitely presented group is hyperbolic iff $\hinf^2(X,\R)$ ``vanishes strongly"; this means that there is a constant~$K>0$ so that for every $f\in \zinf^2(X,\R)$ there is a $c\in\cinf^1(X,\R)$ so that $f=\delta c$ and so that $|c|_\infty\le K|f|_\infty$. \proclaim{Lemma 4.6} The vanishing of $\h1_1$ is equivalent to the linear isoperimetric inequality for summable 1-cycles. Formally, $\h1_1=0$ iff $\exists K>0\ \forall \epsilon>0\ \forall z\in \z1_1\ \exists c_\e\in \c1_2$ such that \roster \item $\hat\partial c_\e=z$, and \item $|c_\e|_1\le K|z|_1+\e$. \endroster \endproclaim \demo{Proof} One direction is clear, since the linear isoperimetric inequality for summable 1-cycles implies \afortiori\ that they admit summable fillings. For the converse, assume that $\h1_1=0$. It is obvious that the sequence $$ \c1_2 @>{\hat\partial}>>\z1_1\to\h1_1\to0 $$ is an exact sequence of linear maps between vector spaces. Because $\h1_1=0$, we have an algebraic isomorphism $\c1_2/\text{Ker}(\hat\partial)\cong\z1_1$. Since $\hat\partial$ is continuous, the Banach inversion theorem implies that the inverse of this bijection is also continuous, so the $\ell_1$ and filling norms on $\z1_1=\b1_1$ are equivalent. This is just the linear isoperimetric inequality for summable 1-cycles. The assertions~(1) and~(2) in the proposition amount to interpreting this in terms of the definition of the filling norm as a quotient norm (i.e., in terms of infima), and the proof is complete. \enddemo Suppose now that $f\in \zinf^2$. If $z\in Z_1$, we set $\=\$, where $c\in \c1_2$ is such that $\partial c=z$. Note that $X$ is contractible, so such $c$ certainly exist. The main step of the argument is contained in the next result. \proclaim{Lemma 4.7} The map~$F:Z_1\to \R$ is well-defined, linear, and $|F|_\infty \le K|f|_\infty$, where $K$ is the constant in Lemma~4.6. \endproclaim \demo{Proof} Let $c,c'\in\c1_2$ be such that $\partial c=\partial c'=z$. Then $c-c'\in \z1_2$. Since $\z1_2=\hat Z_2$ it follows that there is a sequence of elements~$z_n\in Z_2$, $n\ge 1$, so that $z_n\to c-c'$, where convergence is in the sense of the $\ell_1$-norm. Since $Z_2=B_2$, there are elements $y_n\in C_3$ so that $z_n=\partial y_n$ for all $n$. Now calculate $\= \ =\lim \=\lim \=\lim\<\delta f,y_n\>=0$; the second equality holds here since $f\in\zinf^2$, the convergence $z_n\to c-c'$ is in the $\ell_1$-sense, and $\ell_\infty$ is the dual of $\ell_1$. Thus $\=\$, and it follows that $F$ is well-defined. A familiar argument we omit shows that $F$ is linear. Now with $z\in Z_1$, let $\e>0$ and let $c_\e\in \c1_2$ be such that $\partial c_\e=z$ and $|c_\e|_1\le K|z|_1+\e$; the existence of $c_\e$ is guaranteed by Lemma~4.6. Now calculate $|\|=|\|\le |f|_\infty|c_\e|_1 \le |f|_\infty(K|z|_1+\e)$. Since this holds for all~$\e>0$, it follows that $|\|\le K|f|_\infty|z|_1$. It follows that $|F|_\infty\le K|f|_\infty$, and the proof of the lemma is complete. \enddemo Since $F:Z_1\to \R$ is a bounded linear functional, it follows from the Hahn-Banach theorem that $F$ admits a bounded linear extension $H$ to $C_1$ with the same norm. Hence $|H|_\infty\le K|f|_\infty$. Now calculate for $c\in C_2$, $\<\delta H,c\>=\=\=\$, and hence $\delta H=f$. This establishes strong vanishing of $\hinf^2(X,\R)$, and it follows that $G$ is hyperbolic. This completes the proof of Theorem~4.5. \enddemo \proclaim{Corollary 4.8} A finitely presented group~$G$ is hyperbolic iff $\h1_1=\bar\h1_2=0$.\qed \endproclaim \demo{Remark} S. Weinberger has conjectured that a finitely presented group is hyperbolic iff $\h1_1=0$. This assertion remains open at the time of writing. To establish it, in view of Theorem~4.5, would be equivalent to showing that summable 2-cycles can be approximated by 2-cycles of compact support if $\h1_1=0$. \enddemo \def\i1{I^{(1)}} \demo{Remark} The exact sequence (4.2.1) has a higher dimensional analog proved in the same way when $G$ is of type~$\Cal F_{n+1}$, namely, the exact sequence $$ 0@>>> \bar \h1_{n+1}@>>> \hat B_{n}@>\hat\imath>>\z1_{n} @>>>\h1_{n} @>>>0. $$ To understand this better, one introduces the group $\i1_n=\hat Z_n/\b1_n$ which is defined for groups~$G$ of type~$\Cal F_{n+1}$. Then one has from the filtration $\b1_n\subset \hat Z_n\subset \z1_n$ the short exact sequence $$ 0\to \i1_n\to \h1_n\to \bar\h1_n\to 0, $$ which is the analog of (4.3.1), and the exact sequence $$ 0@>>> \bar\h1_{n+1}@>>> \hat B_n@>\hat\imath>>\hat Z_n@>>> \i1_n@>>>0. $$ $\i1_n$ should be thought of as the obstruction to linear filling of summable $n$-cycles while $\bar \h1_n$ is the obstruction to approximating summable $n$-cycles by $n$-cycles of compact support. Thus Corollary~4.8 says that the finitely presented group~$G$ is hyperbolic iff both obstructions $\i1_1$ and $\bar \h1_2$ vanish. \enddemo \subhead{\S 5. Vanishing theorems for $\h1_2$} \endsubhead Proposition~4.4 established that $\h1_2$ vanishes for a hyperbolic group. In this section we prove that for all 1-relator groups and for all fundamental groups of finite piecewise Euclidean 2-complexes of nonpositive curvature one has vanishing of $\h1_2$. For these groups the hyperbolicity criterion of Corollary~4.8 reduces to the vanishing of~$\h1_1$. First let $G$ be a 1-relator group, so $G$ is defined by a presentation consisting of a finite set of generators and a single defining relation. We shall prove \proclaim{Theorem 5.1} If $G$ is a 1-relator group then $\h1_2(G,\R)=0$. \endproclaim An immediate consequence of this result is \proclaim{Corollary 5.2} The 1-relator group~$G$ is hyperbolic iff $\h1_1(G,\R)=0$. \endproclaim \demo{Proof} If $G$ has nontrivial torsion, then it is known that $G$ is hyperbolic; this is a consequence of the ``spelling theorem" of B.~B.~Newman \cite{LS} p.~205. Thus $\h1_1$ and $\h1_2$ are both zero in this case. If $G$ is torsion-free, then by Lyndon's identity theorem \cite{LS} the 1-relator presentation for $G$ is aspherical, so hyperbolicity of $G$ is equivalent to the vanishing of $\h1_1$ and $\h1_2$ by Corollary~4.3. But $\h1_2$ vanishes by Theorem~5.1, so hyperbolicity is equivalent to the vanishing of $\h1_1$, and the corollary follows from the theorem. \enddemo \demo{Definition} Suppose that $K$ is a subcomplex of the CW complex~$L$ such that $K=T\times I$, where $T$ is a complex and $I=[0,1] \subset\R$. We say that $K$ is {\it isolated\/} in $L$ if $L-(T\times(0,1))$ is a subcomplex of~$L$. \enddemo \proclaim{Lemma 5.3} Suppose that the subcomplex $K=T\times I$ with $T$ a tree is isolated in the 2-complex~$L$. If $f$ is an $\ell_1$ 2-cycle on $L$, then $f$ vanishes on all 2-cells of $L$ of the form $e\times(0,1)$, where $e$ is an edge of $T$. \endproclaim \demo{Proof} If $e$ is an edge of $T$, we define $F(e)=f(e\times(0,1))$. It is readily checked that $F$ is a summable 1-cycle on $T$. But $\z1_1(T,\R)=0$ for every tree~$T$, by Corollary~3.4. It follows that $F=0$, and hence $f(e\times(0,1))=0$ for all edges of $T$, and the lemma is established. \enddemo \proclaim{Lemma 5.4} Suppose that $G=H*_F K$ {\rm(}resp. $G=H*_F${\rm)} where $F$ is a finitely generated free group and $H$ and $K$ {\rm(}resp. $H${\rm)} admit finite aspherical presentations. Then $\h1_2(G,\R)=0$ iff $\h1_2(H,\R)=0=\h1_2(K,\R)$ {\rm(}resp. $\h1_2(H,\R)=0${\rm)}. \endproclaim \demo{Proof} We write out the proof in the HNN case, $G=H*_F$; the amalgam case involves only notational changes. Let $X'$ be a finite 2-dimensional $K(H,1)$. Then we can construct a $K(G,1)$ by taking $Y'=X'\cup (\Gamma\times [-1,2])/\sim$, where $\Gamma$ is a finite connected graph, and $\Gamma\times\{i\}$ is attached to $X'$ by cellular maps inducing the given injective homomorphisms $F\to H$, $i=-1, 2$. We can subdivide the interval $[-1,2]$ at points $0,1$ and take the induced product cell structure on $\G\times [-1,2]$, so that $\G\times[0,1]$ becomes a subcomplex of $Y'$. The proof that $Y'$ is a $K(G,1)$ is standard and makes use of the Mayer-Vietoris exact sequence in homology for $Y$, the universal covering space of $Y'$. Note that the preimage of $\G\times I$ in $Y$ is a disjoint union of isolated subcomplexes, each isomorphic to $T\times I$, where $T$ is the universal covering of~$\G$. Furthermore, the preimage of $X'$ in $Y$ is a disjoint union of isomorphic copies of $X$, the universal covering space of~$X'$. We suppose first that $\h1_2(G,\R)=0$, and we let $f$ be a summable 2-cycle on $X$. Then $f$ can be extended to a summable 2-cycle~$F$ on $Y$ by defining $F$ to be $f$ on one connected component of the preimage of $X'$ in $Y$ and zero on all of the other such connected components, as well as setting $F=0$ on all 2-cells of $Y$ over those of $\G\times (-1,2)$. But $\h1_2(Y,\R)=\z1_2(Y,\R)=0$, so it follows that $F=0$. Thus $f=0$ as well. Next suppose that $\h1_2(H,\R)=0$ and let $F$ be a summable 2-cycle on $Y$. By Lemma~5.3 it follows that $F$ vanishes on all 2-cells of $Y$ which map to those of $\G\times(-1,2)$. But this means that $F$ is supported on the preimage of $X'$ in $Y$. It follows from $\h1_2(H,\R)=\z1_2(X,\R)=0$ that $F$ restricted to each connected component of the preimage of $X'$ in $Y$ is zero, and hence $F=0$. This completes the proof. \enddemo \demo{Proof of Theorem~5.1} It suffices to consider torsion-free 1-relator groups, as the argument in the first paragraph of the proof of Corollary~5.2 shows. Given a 1-relator presentation~$\PP$ whose relator~$R$ is cyclically reduced and not a proper power in the free group of the generators, we shall prove that $\h1_2(G,\R)=0$, where $G$ is the group defined by $\PP$, by induction on the length~$\ell(R)$ of the word~$R$. The induction starts when $\ell(R)=1$, for in this case $G$ is free. For the inductive step, we need to recall the structure of 1-relator groups in the form presented in \cite{MS} (\cf also \cite{LS} pages~198--200). We assume that $G$ has a 1-relator presentation~$\PP$ with $\ell(R)=n>1$. There are two cases, depending on whether some generator of $\PP$ actually occurring in $R$ has exponent sum~0, or the contrary case when none of the generators appearing in $R$ has exponent sum~0. In the first case $G=H*_F$ where $H$ is a 1-relator group defined by a 1-relator presentation whose defining relator~$S$ has $\ell(S)0$ so that $K=G*_{g=x^d}\$, where $K=H*_F$; here $H$ admits a 1-relator presentation whose defining relator~$S$ has $\ell(S)$, $p,q>0$. One might ask whether the homological methods introduced here can be of use in attacking this previously unapproachable question. \enddemo \proclaim{Theorem 5.6} If $G$ is the fundamental group of a finite piecewise Euclidean simplicial 2-complex~$K$ of nonpositive curvature, then $\h1_2(G,\R)=0$. \endproclaim \demo{Proof} For the definitions and properties of complexes of nonpositive curvature and CAT(0) spaces we refer the reader to W. Ballmann's article in \cite{GH}. We need here the fact that the universal cover~$\wt K$ of $K$ is a simply connected nonpositively curved complex with finitely many isometry types of simplices, and hence, by the theorem of M. Bridson \cite{Br}, $\wt K$ possesses unique geodesic segments connecting any pair of points. We need \proclaim{Lemma 5.7} Let $\sigma$ be an open 2-simplex of $\wt K$ and let $p\in \sigma$. Then there is a direction~$u$ in the tangent space of $\sigma$ at $p$ so that no geodesic segment in $\wt K$ passing through $p$ with tangent directions $\pm u$ passes through a vertex of $\wt K$. \endproclaim \demo{Proof} There are only countably many vertices $v\in \wt K\sk 0$ so there are only countably many tangent directions at $p$ of the geodesic segments $[p,v]$. Choose $u$ to avoid all of these directions and their negatives. \enddemo Returning to the proof of the theorem, let $f$ be a summable 2-cycle on $\wt K$ and let $\sigma$ be an open 2-simplex of $\wt K$. Let $p\in \sigma$ and choose the direction~$u$ in the tangent space to $\sigma$ at $p$ as in the lemma. Consider $T$, the union of all geodesic segments through~$p$ in $\wt K$ whose tangent directions at $p$ are $\pm u$. Then $T$ is a tree with a discrete set of vertices where $T$ intersects the 1-skeleton of $\wt K$. The edges of $T$ are the nonempty intersections of $T$ with 2-simplices of $\wt K$. Such an edge~$e$ has a product neighborhood in the 2-simplex~$\sigma$ in which it lies, and these product neighborhoods fit together to give a product neighborhood $N=T\times I$ of $T$ in $\wt K$ which does not meet $\wt K\sk 0$ (note however that one cannot choose $N$ of uniform thickness in the normal $I$-direction in general in the metric of $\wt K$). Having fixed an orientation on $I$, giving an orientation on $N\cap\sigma \cong e\times \supo I$ is equivalent to giving an orientation on the edge~$e$. In this way we define a 1-cochain~$F$ on $T$ so that $F(e)=f(\sigma)$, where $e=T\cap \sigma$ and where the orientations are compatible in the sense just described. That $F$ is a summable 1-chain on $T$ follows from the summability of $f$ and that $F$ is summable 1-cycle follows from the facts that $f$ is a summable 2-cycle and that $T$ does not meet $\wt K\sk 0$. It follows from Corollary~3.4 that $F=0$, and hence in particular $f(\sigma)=0$. Since $\sigma$ was an arbitrary 2-simplex, it follows that $f=0$, and the proof is complete. \enddemo \demo{Remark} For higher dimensional CAT(0) spaces it is certainly not the case that $\h1_2$ vanishes (although the question of approximating summable 2-cycles by 2-cycles of compact support remains open). For example, if $X$ is the tesselation of $\R^3$ by the unit cube lattice and $z_n$ is the 2-cycle which is the boundary of a cube of side~$n$ with the orientation determined by the outward pointing normal, then it is easy to see that $\sum_{n\ge 1}\dfrac{z_n}{n^4}$ is a summable 2-cycle which admits no summable filling; thus $\h1_2(\Z^3,\R)\ne 0$. \enddemo \proclaim{Corollary 5.7} The fundamental group~$G$ of a finite piecewise Euclidean 2-complex of nonpositive curvature is hyperbolic iff $\h1_1(G,\R)=0$.\qed \endproclaim \subhead{\S 6. Examples} \endsubhead Theorem~3.3 says that if $\G$ is a graph, then elements of $\text{Ker}(\hat \partial)$ can be $\ell_1$-approximated by elements of $\text{Ker}(\partial)$, where $\partial:C_1(\G,\R)\to C_0(\G,\R)$ is the boundary map and $\hat\partial $ is its $\ell_1$-completion. Equivalently, $\z1_1(X,\R)=\hat Z_1(X,\R)$ for every CW-complex~$X$. It is natural to ask whether there is an analogous approximation theorem for $n$-cycles, \ie\ whether $\z1_n(X,\R)=\hat Z_n(X,\R)$ for $n\ge 2$. The next example shows that it is futile to look for general results of this type. % for $n\ge 3$. \demo{Example 6.1 {\rm (E. Formanek% \footnote{private communication}% )}} Let $G$ be the free group with free basis~$\{x,y\}$. Then the row vector $M=\pmatrix 2-x, y-2\endpmatrix$ defines a monomorphism of right $\R G$-modules $(\R G)^2\to \R G$ by $\pmatrix a\\b\endpmatrix\mapsto (2-x)a+(y-2)b$. However if $a=1+\frac12x+\frac 14x^2+\dots$ and $b=1+\frac 12y+\frac 14y^2+\dots$, then $(2-x)a=(2-y)b=2$, so the induced map of $\ell_1$-completions $\widehat {\R G}^2\to \widehat{\R G}$ is not injective. For each $n\geq2$ one may use this to build a CW complex $X$ of dimension $n$ and a summable $n$-chain thereon that cannot be $\ell_1$-approximated by compactly supported $n$-chains. Furthermore, we may take $X$ to admit a cocompact free action of $G$ and to be simply connected (if $n>2$) or aspherical (if $n=2$). The topological construction is standard: if $n>2$ then take $X$ as the universal cover of $X'$, where $X'$ is the join of two circles and an $n-1$-sphere with a pair of $n$-cells attached in such a way that boundaries of their lifts to $X$ are given by the entries of Formanek's matrix~$M$. Then $M$ gives the boundary map $C_n\to C_{n-1}$, so $Z_n(X)=0$ but $\z1_n(X)\neq0$. The same basic construction yields a slightly different result for $n=2$. Then $X'$ may be taken to be the $2$-complex associated to the presentation $$ \langle x,y,z\mid 1=z^2xz^{-1}x^{-1}=z^2yz^{-1}y^{-1}\rangle $$ and $X$ to be the covering space of $X'$ associated to the normal closure of $z$; the free group $G$ is the group of covering transformations. $X'$ is aspherical because it is the presentation complex of an iterated HNN extension of $\langle z\rangle\cong\Z$. The boundary map $C_2(X)\to C_1(X)$ has image in a rank one $G$-submodule of $C_1(X)$, with matrix given by $M$. As before, this shows that $Z_2(X)=0$ but $\z1_2(X)\neq0$. %For each $n\geq3$ one can build a %a finite complex $X_n'$ with fundamental group~$G$ %and universal cover~$X_n$ %such that the boundary map %$C_n(X_n,\R)\to C_{n-1}(X_n,\R)$ is given by %the matrix~$M$. %Since $Z_n(X_n,\R)=0$, it follows %that $\z1_n(X_n,\R)\ne \hat Z_n(X_n,\R)=0$. \enddemo \demo{Question} If $X$ is a simply-connected 2-complex which admits a discrete cocompact group action by cellular homeomorphisms, can summable 2-cycles on $X$ be approximated by 2-cycles of compact support? This is for our purposes the most interesting open case of the general question of approximating summable $n$-cycles by cycles of compact support that is not already answered positively by Theorem~3.3 for $n=1$ and negatively by examples constructed from Formanek's example in the preceding paragraph. \enddemo In the introduction we noted that the lack of reflexivity of the Banach space~$\ell_1$ has the peculiar effect that the vanishing theorems for hyperbolic groups are in different degrees, $\hinf^2$ and $\h1_1$. We complete this discussion by showing \proclaim{Proposition 6.2} If $G$ is an infinite finitely generated group then $\hinf^1(G,\R)$ and the reduced group $\bar\hinf^1(G,\R)$ are both nonzero. \endproclaim \demo{Proof} Let $X'$ be a $K(G,1)$ with finite 1-skeleton and let $X$ be the universal cover of $X'$. Choose a vertex~$v_0$ of $X$ as base point and give the one-skeleton $X\sk 1$ the path metric~$d$ where each edge has length~1. For an oriented edge~$e$ of $X$ set $\D(e)=d(v_0,\tau e)-d(v_0,\iota e)$. Then $\D$ is a 1-cocycle, and the triangle inequality %(\cf \cite{Ge3}). shows that $|\D|_\infty=1$ (where $|\D|_\infty$ denotes the sup (or $\ell_\infty$) norm of the cochain~$\D$). Furthermore $\D\notin \binf^1(X,\R)$ since its potential function $D:X\sk 0\to \R$, given by $D(v)=d(v_0,v)$, is unbounded, as follows since $G$ is infinite. As for the reduced cohomology group $\bar\hinf^1(X,\R)$, recall that this is the quotient of $\zinf^1(X,\R)$ by the closure of $\binf^1(X,\R)$. We claim that the cocycle $\D$ is not in the closure of $\binf^1(X,\R)$. To see this we need the following \proclaim{Lemma 6.3} If $f\in \zinf^1(X,\R)$ is in the closure of $\binf^1(X,\R)$, then every potential function~$F$ for $f$ satisfies $$ \lim_{d(v_0,v)\to \infty} \frac{F(v)}{d(v_0,v)}=0. $$ \endproclaim \demo{Proof} Let $\epsilon>0$ and let $h\in \cinf^0(X,\R)$ be such that $|f-\delta h|_\infty\le \e$. Let $\g=e_1e_2\dots e_n$ be a geodesic edge-path starting at $v_0$ and ending at $v$. Then $$ F(v)-F(v_0)-(h(v)-h(v_0))=\sum_{i=1}^n( f(e_i)-\delta h(e_i)), $$ so $|F(v)|\le n\e +|F(v_0)|+2|h|_\infty$, and hence $\dfrac {|F(v)|}n \le 2\e$ for $n$ sufficiently large. \enddemo Returning to the proof of the Proposition we see that $\dfrac {D(v)}{d(v_0,v)}=1$ for all vertices~$v$, so it follows from Lemma~6.3 that $\D$ is not in the closure of $\binf^1(X,\R)$. Hence $\bar \hinf^1(X,\R)\ne 0$. \enddemo \subhead{Appendix} \endsubhead Let $\G$ be a graph and let $A$ be a normed abelian group. It is an open question whether summable 1-cycles on $\G$ with coefficients in $A$ can be approximated by cycles of compact support in general, although the case $A=\R$ was settled affirmatively by Theorem~3.3. Another special case will be established in this Appendix. We recall that $A$ is called {\it ultrametric\/} if $|a+b|\le\max(|a|,|b|)$ for all $a,b\in A$. We call the norm {\it discrete\/} if the set $\{|a|, a\in A\}$ has only 0 as a limit point. For example the $p$-adic completion of the rational numbers is ultrametric with discrete norm. \proclaim{Theorem A1} Let $\G$ be a graph and let $A$ be a ultrametric normed abelian group with discrete norm. Then summable 1-cycles on $\G$ with coefficients in $A$ can be approximated in the $\ell_1$-sense by 1-cycles of compact support. \endproclaim \demo{Proof} Let $f$ be a nonzero summable 1-cycle on $\G$ with coefficients in $A$. It follows from summability that for each $r>0$ the set $E_r$ of edges in $\G$ so that $|f(e)|\ge r$ is finite, and consequently $f$ is nonzero on at most a countable set of edges. Let $M$ be the maximum value of $|f(e)|$ as $e$ runs over the edges of $\G$ and let $|f(e_1)|=M$. We need \proclaim{Lemma A2} There exists a simple circuit $z_1$ so that $|f(e )|=M$ for each edge~$e$ in $z_1$. \endproclaim \demo{Proof} If $v=\tau e_1$, then $\sum_{\tau e=v}f(e)=0$ from the cycle condition, and the term $f(e_1)$ in the sum is of maximal norm. It follows from the ultrametric inequality that there exists an edge $ e_2$ with $\iota e_2=v$ and so that $|f(e_2)|=M$. Continuing in this way, we produce by induction a path $e_1e_2\dots e_n$ for each $n\ge 1$ so that $|f(e_i)|=M$. Since there are only finitely many edges~$e$ altogether with $|f(e)|=M$, there must be exist a number $k>0$ so that $e_i=e_{i+k}$ for some $i\ge 1$. Taking $k$ minimal produces a simple circuit~$z_1$ satisfying the conclusion of the lemma. \enddemo Next let $f_1=f-a_1z_1$, where $a_1$ is the value of $f$ on one of the edges of $z_1$ (it makes no difference for the argument which edge is chosen). It follows from the ultrametric inequality that $|f_1(e)|\le |f(e)|$ for each edge~$e$ and consequently $|f_1|_1\le |f|_1$, where $|f|_1$ denotes the $\ell_1$-norm of $f$, which, we recall from (2.1) above, is the sum of the norms of the values of $f$ on an orientation~$\Cal O$ of the edges of $\G$. Also note that $f_1$ has at least one fewer edge than~$f$ with value of maximal norm. Now repeat the process, replacing $f$ by $f_1$, finding a simple edge-circuit~$z_2$ of edges on which $f_1$ assumes values of maximal norm, defining $f_3=f_2-a_2z_2$, where $a_2$ is value of $f_1$ on one of the edges of $z_2$, and so forth, producing thereby the sequence of summable 1-cycles $f_2,f_3,\dots$. The relevant facts about this sequence are \roster \item $|f_{n+1}(e)|\le |f_n(e)|$ for every edge~$e$ of $\G$, and $|f_{n+1}(e)|<|f_n(e)|$ for at least one edge~$e$ on which $f_n$ assumes its value of maximal norm, and \item $|f_{n+1}|_1\le |f_n|_1\le |f|_1$ for all $n\ge 1$. \endroster It follows from (1) and the discreteness of the norm on $A$ that $\lim_{n\to\infty}|f_n(e)|=0$ for each edge~$e$ while $|f_n|_1\le |f|_1$ for all $n\ge 1$. But the Lebesgue dominated convergence theorem (applied to the sequence of real valued functions $g_n$ on the discrete measure space of edges of $\G$, where $g_n$ is given by $g_n(e)=|f_n(e)|$) implies that $\lim_{n\to\infty}|f_n|_1=|\lim_{n\to\infty} g_n|_1=0$. If we set $s_n=a_1z_1+a_2z_2+\dots+a_nz_n$, then $f=s_n+f_n$ for each $n\ge 1$, $s_n$ is a 1-cycle of compact support, and $\lim_{n\to \infty}s_n=f$ where the limit is taken in the sense of the $\ell_1$-norm. Thus $f$ can be approximated by 1-cycles of compact support, and the proof of the theorem is complete. \enddemo \demo{Remark} It follows from arguments of \cite{Mi} that if the graph~$\G$ is combable in the sense of \cite{EC}, then approximation of summable 1-cycles by cycles of compact support is possible for all normed abelian coefficient groups. Furthermore, it is a result of \cite{Fl} that a summable 1-cycle~$f$ with values in $A$ on the connected graph~$\G$ satisfying the stronger assumption that $\sum_{e}(d(v_0,\iota e)+1)|f(e)|<\infty$, where $v_0$ is the base point and $d$ is the path metric assigning length~1 to each edge, can be approximated by cycles of compact support. It is also not difficult to see that a summable 1-cycle on a tree must necessarily be zero. 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