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 $\Sub(F\wrΓ)$ supported on ${\oplus }_{\Gamma }F$ also form a Poulsen simplex.

This version has new results: the determination of the perfect kernel of the space of subgroups and its Cantor-Bendixon rank and more general results on lamplighters

References in corpus (2)

• Kesten's theorem for Invariant Random Subgroups
• On the lattice of subgroups of the lamplighter group

Cited by in corpus (6)

• Furstenberg entropy realizations for virtually free groups and lamplighter groups
• Unimodular measures on the space of all Riemannian manifolds
• Invariant random subgroups of linear groups
• Unimodularity of Invariant Random Subgroups
• Characteristic random subgroups of geometric groups and free abelian groups of infinite rank
• Generic IRS in free groups, after Bowen