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 of Brooks for the spectrum of the Laplacian on covers of Riemannian manifolds.