Isoperimetric inequalities and the homology of groups

An isoperimetric function for a finitely presented group G bounds the number of defining relations needed to prove that a word w =G 1 in terms of the length of w. Suppose G = F/R is a finitely presented group where F is a finitely generated free group and R is the normal closure of a finite set of relators. For each w ∈ R we define the area ∆(w) to be the least number of uses of the given relators needed to prove that w =G 1. The isoperimetric function ΦG(n) is defined to be the maximum of the areas of all the words in R of length ≤ n. One compares the size of such functions using a suitable order 1 on functions. Different presentations give rise to ' equivalent isoperimetric functions where ' is the equivalence relation determined by 1 . Interest in such functions was stimulated by Gromov [13] who characterized word hyperbolic groups as those with linear isoperimetric function, that is, ΦG(n) ' n. Further information was obtained by Cannon et al [8] who showed that automatic groups satisfy ΦG 1 n and that asynchronously automatic groups satisfy ΦG 1 2. They also showed the free nilpotent group of class 2 and rank 2 has ΦG(n) ' n. Gersten [10] used geometric arguments to obtain interesting lower bounds of exponential type for certain one relator groups. He exploited the fact that, for a torsion-free one relator groupG, the usual 2 dimensional complex having fundamental group isomorphic to G is aspherical. The general technique for giving upper bounds for ΦG(n) is to describe a solution to the word problem for G and analyze its use of defining relations. It seems quite difficult to establish lower bounds for ΦG(n) in this manner. In this paper we begin a more algebraic study of isoperimetric inequalities and introduce new methods for establishing lower bounds for ΦG. Instead of trying to analyze R directly, we consider some of its quotient groups for which additional machinery is available. For instance, we consider for each w ∈ R, its image w[R,F ] in R/[R,F ] and define a suitable “centralized” isoperimetric function Φ G (n) such that Φ G (n) 1 ΦG(n). Using the Hopf formula it can be shown that R/[R,F ] is a direct sum of the second homology group H2(G,Z) with a free abelian