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.