Critical exponents of invariant random subgroups in negative curvature

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 ∂ 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) > 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.