Invariant random subgroups of acylindrically hyperbolic groups
Also appears in collection : GAGTA-9: geometric, asymptotic and combinatorial group theory and applications / GAGTA-9 : Théorie géométrique, asymptotique et combinatoire des groupes et applications
A subgroup $H$ of an acylindrically hyperbolic groups $G$ is called geometrically dense if for every non-elementary acylindrical action of $G$ on a hyperbolic space, the limit sets of $G$ and $H$ coincide. We prove that for every ergodic measure preserving action of a countable acylindrically hyperbolic group $G$ on a Borel probability space, either the stabilizer of almost every point is geometrically dense in $G$, or the action is essentially almost free (i.e., the stabilizers are finite). Various corollaries and generalizations of this result will be discussed.
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