Amenability, critical exponents of subgroups and growth of closed geodesics (Q303613)

Let \(\Gamma\) be a (non-elementary) convex co-compact group of isometries of a pinched Hadamard manifold \(X\). The authors 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. They also show a similar result for the exponential growth rate of closed geodesics on \(X/\Gamma\). These statements are analogues of classical results of \textit{H. Kesten} [Math. Scand. 7, 146--156 (1959; Zbl 0092.26704)] for random walks on groups and \textit{R. Brooks} [J. Reine Angew. Math. 357, 101--114 (1985; Zbl 0553.53027)] for the spectrum of the Laplacian on covers of Riemannian manifolds.