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

17 pages

References in corpus (1)

• An extension of Kesten's criterion for amenability to topological Markov chains

Cited by in corpus (1)

• Growth gap in hyperbolic groups and amenability