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 $${\delta(\mu)}$$ of any discrete invariant random subgroup $${\mu}$$ of the locally compact group G and show that $${\delta(\mu) > \frac{d}{2}}$$ in general and that $${\delta(\mu) = d}$$ if $${\mu}$$ 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.