Critical exponents of invariant random subgroups in negative curvature

TL;DR

This paper investigates the critical exponents of invariant random subgroups in negatively curved spaces, establishing bounds and conditions under which these subgroups are lattices, using ergodic theory and hyperbolic group actions.

Contribution

It introduces a definition of critical exponents for invariant random subgroups in hyperbolic spaces and proves new bounds and characterizations, including when such subgroups are lattices.

Findings

Critical exponent exceeds half the boundary dimension in general.

Equal to the boundary dimension for divergence type subgroups.

Invariant random subgroups of divergence type are lattices in rank-one Lie groups with property (T).

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.