Faithful invariant random subgroups in acylindrically hyperbolic groups

Abstract

Building on work from Sun and Kechris-Quorning, we prove that every acylindrically hyperbolic group 𝐺 admits a weakly mixing probability measure preserving action 𝐺 ↷ (𝑋, , 𝜇) which is faithful but not essentially free. In other words, 𝐺 admits a weakly mixing nontrivial faithful IRS. We also prove that every nonelementary hyperbolic group admits a characteristic random subgroup with the same properties.