Realizing invariant random subgroups as stabilizer distributions

Abstract

Suppose that $\nu$ is an ergodic invariant random subgroup of a countable group $G$ such that $[N_G(H):H]=n<\infty$ for $\nu$-a.e. $H\in \mathrm{Sub}$. In this paper, we consider the question of whether $\nu$ can be realized as the stabilizer distribution of an ergodic action $G\curvearrowright (X,\mu )$ on a standard Borel probability space such that the stabilizer map $x\mapsto G_x$ is $n$-to-one.