Invariant random subgroups of the lamplighter group

Abstract

Let $G$ be one of the lamplighter groups $({\mathbb{Z}/p\bz})^n\wr\mathbb{Z}$ and $\Sub(G)$ the space of all subgroups of $G$. We determine the perfect kernel and Cantor-Bendixson rank of $\Sub(G)$. The space of all conjugation-invariant Borel probability measures on $\Sub(G)$ is a simplex. We show that this simplex has a canonical Poulsen subsimplex whose complement has only a countable number of extreme points. If $F$ is a finite group and $\Gamma$ an infinite group which does not have property $(T)$ then the conjugation-invariant probability measures on $\Sub(F\wr\Gamma)$ supported on $\oplus_\Gamma F$ also form a Poulsen simplex.