Invariant random subgroups of the lamplighter group

2012/06/28 by Lewis Bowen, Bowen, Lewis, Rostislav Grigorchuk +3
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1206.6780

openalex publication_date 2012/06/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be one of the lamplighter groups (ℤ/p\bz)ⁿ\wrℤ 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 Γ an infinite group which does not have property (T) then the conjugation-invariant probability measures on \Sub(F\wrΓ) supported on ⊕_ΓF also form a Poulsen simplex.

Citations

• Random walks on random coset spaces with applications to Furstenberg\n entropy

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