Invariant random subgroups of groups acting on hyperbolic spaces

Osin, D.

doi:10.1090/proc/13469

Invariant random subgroups of groups acting on hyperbolic spaces
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Proc. Amer. Math. Soc. 145 (2017), 3279-3288 Request permission

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 geometrically dense in $G$ if the limit sets of $H$ and $G$ on $\partial S$ 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.

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