A note on curvature and fundamental group

Define the growth function γ associated with a finitely generated group and a specified choice of generators {gl7 -, gp} for the group as follows (compare [9]). For each positive integer s let γ(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 γ(s) = 2s + 2s -f 1.) We will see that the asympotic behavior of γ(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. Gύnther [3], to prove two theorems. Theorem 1. // M is a complete n-dimensional Riemannian manifold whose mean curvature tensor R^ is everywhere positive semidefinite, then the growth function γ(s) associated with any finitely generated subgroup of the fundamental group 7ΓjM must satisfy