A note on curvature and fundamental group

Define the growth function associated with a finitely generated group and a specified choice of generators g l7 -, g p for the group as follows (compare 9). For each positive integer s let (s) be the number of distinct group elements which can be expressed as words of length < s in the specified generators and their inverses. (For example, if the group is free abelian of rank 2 with specified generators x and y, then (s) = 2s 2 + 2s -f 1. ) We will see that the asympotic behavior of (s) as s - oo is, to a certain extent, independent of the particular choice of generators (Lemma 1). This note will make use of inequalities relating curvature and volume, due to R. L. Bishop 1, 2 and P. Gnther 3, to prove two theorems.