A note on curvature and fundamental group

Abstract

Define the growth function γ 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 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 2 + 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 7Γj M must satisfy γ(s) < constant s n .It is conjectured that the group π x M itself must be finitely generated.The constant in this inequality will depend, of course, on the particular set of generators which is used to define γ(s).Theorem 2. // M is compact Riemannian with all sectional curvatures less than zero, then the growth function of the fundamental group π^M is at least exponential: γ(s) > a°f or some constant a > 1.In both cases, any set of generators for the group may be used in defining Remarks.Note that there is always an exponential upper bound for γ(s).In fact the inequality