On an isoperimetric inequality for in finite finitely generated groups

Let Gamma be an infinite, finitely generated group. We prove that for any finite subset A of Gamma the following inequality is true: \ A \ less than or equal to Sigma(gamma epsilon partial derivative A) dist(e,gamma), where dist (e, gamma) is a distance in Gamma of gamma to the identity element e, and partial derivative A is a boundary of A. This inequality implies that the volume form on the universal cover of a compact Riemannian manifold with infinite fundamental group has a primitive of at most linear growth. (C) 2000 Elsevier Science Ltd. All rights reserved.