Invariant random subgroups of groups acting on hyperbolic spaces

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.Comment: Minor corrections. To appear in Proc. AM

Acylindrically hyperbolic groups

We say that a group $G$ is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that the class of acylindrically hyperbolic groups coincides with many other classes studied in the literature, e.g., the class $C_{geom}$ introduced by Hamenstadt, the class of groups admitting a non-elementary weakly properly discontinuous action on a hyperbolic space in the sense of Bestvina and Fujiwara, and the class of groups with hyperbolically embedded subgroups studied by Dahmani, Guirardel, and the author. We also record some basic results about acylindrically hyperbolic groups for future use.Comment: Lemma 6.8 is adde