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\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. However the general
question, of approximating summable 1-cycles on arbitrary
graphs with values in an arbitrary normed abelian group
by 1-cycles of compact support, remains open.
\enddemo
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\end