Amenability,critical exponents of subgroups and growth of closed geodesics

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.