Amenability, critical exponents of subgroups and growth of closed geodesics

Abstract

Let $$\Gamma $$ be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold X. We show that a normal subgroup $$\Gamma _0$$ has critical exponent equal to the critical exponent of $$\Gamma $$ if and only if $$\Gamma /\Gamma _0$$ is amenable. We prove a similar result for the exponential growth rate of closed geodesics on $$X/\Gamma $$ . These statements are analogues of classical results of Kesten for random walks on groups and Brooks for the spectrum of the Laplacian on covers of Riemannian manifolds.