Invariant random subgroups of groups acting on hyperbolic spaces

TL;DR

This paper classifies invariant random subgroups of groups acting on hyperbolic spaces, showing they are either geometrically dense or contained in the elliptic radical, with implications for group actions and stabilizer sizes.

Contribution

It establishes a dichotomy for invariant random subgroups of hyperbolic space groups, linking geometric density to subgroup classification and action stabilizer properties.

Findings

Invariant random subgroups are either geometrically dense or elliptic radical-contained.

Ergodic actions have either finite stabilizers or large, hyperbolic stabilizers.

Provides a classification framework for subgroup behavior in hyperbolic group actions.

Abstract

Suppose that a group $G$ acts non-elementarily on a hyperbolic space $S$ and does not fix any point of $\partial S$. A subgroup $H\le G$ is said to be geometrically dense in $G$ if the limit sets of $H$ and $G$ coincide and $H$ does not fix any point of $\partial S$. We prove that every invariant random subgroup of $G$ is either geometrically dense or contained in the elliptic radical (i.e., the maximal normal elliptic subgroup of $G$). In particular, every ergodic measure preserving action of an acylindrically hyperbolic group on a Borel probability space $(X,\mu)$ either has finite stabilizers $\mu$-almost surely or otherwise the stabilizers are very large (in particular, acylindrically hyperbolic) $\mu$-almost surely.