Critical exponents of invariant random subgroups in negative curvature

Abstract

Let $X$ be a proper geodesic Gromov hyperbolic metric space and let $G$ be a cocompact group of isometries of $X$ admitting a uniform lattice. Let $d$ be the Hausdorff dimension of the Gromov boundary $\partial X$. We define the critical exponent $δ(μ)$ of any discrete invariant random subgroup $μ$ of the locally compact group $G$ and show that $δ(μ) > \frac{d}{2}$ in general and that $δ(μ) = d$ if $μ$ is of divergence type. Whenever $G$ is a rank-one simple Lie group with Kazhdan's property $(T)$ it follows that an ergodic invariant random subgroup of divergence type is a lattice. One of our main tools is a maximal ergodic theorem for actions of hyperbolic groups due to Bowen and Nevo.