 
Summary: A HOMOLOGICAL CHARACTERIZATION
OF HYPERBOLIC GROUPS
D. J. Allcock and S. M. Gersten
Abstract. A finitely presented group G is hyperbolic iff H (1)
1
(G; R) = 0 =
¯
H (1)
2
(G; R), where H (1)
\Lambda (resp. ¯
H (1)
\Lambda ) denotes the ` 1 homology (resp. reduced ` 1
homology). If \Gamma is a graph, then every ` 1 1cycle in \Gamma with real coefficients can be
approximated by 1cycles of compact support. A 1relator group G is hyperbolic iff
H (1)
1
(G; R) = 0.
x1. Introduction.
In [Ge1] the second author showed that a finitely presented group G is word
