Invariant random subgroups of groups acting on hyperbolic spaces

Abstract

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