Amenability, Critical Exponents of Subgroups and Growth of Closed Geodesics

Let $\mathrm{\Gamma }$ be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold $X$. We show that a normal subgroup ${\mathrm{\Gamma }}_0$ has critical exponent equal to the critical exponent of $\mathrm{\Gamma }$ if and only if $\mathrm{\Gamma }\mathrm{/}{\mathrm{\Gamma }}_0$ is amenable. We prove a similar result for the exponential growth rate of closed geodesics on $X\mathrm{/}\mathrm{\Gamma }$. 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.