Stabilizer Subgroups and the Simplicity of Reduced Crossed Products

Given a minimal action $G\curvearrowright X$ of a countable group $G$ on a compact space $X$, we prove that if the reduced crossed product $G\ltimes_rC(X)$ is simple, then there exists a point whose stabilizer subgroup has trivial amenable radical. As a consequence, we give a complete characterization of the simplicity of the reduced crossed product of minimal actions of countable linear groups, hyperbolic groups, and, more generally, for groups with countably many amenable subgroups. This answers a question of Ozawa (2014) for these classes of groups. Furthermore, in the case of an infinite uniformly recurrent subgroup of a $C^*$-simple group, we prove that almost every subgroup has a trivial amenable radical, with respect to a fully supported, atomless probability measure.