BOUNDARIES OF DENSE SUBGROUPS OF TOTALLY DISCONNECTED GROUPS

Let Γ be a countable discrete group and let H be a lcsc totally disconnected group, L a compact open subgroup of H, and ρ : Γ → H a homomorphism with dense image. In this paper we construct, for every bi-Linvariant probability measure θ on H, an explicit Furstenberg discretization τ of θ such that the Poisson boundary (B_θ, ν_θ) of (H, θ) is a τ-boundary, where Γ acts on B_θ via the homomorphism ρ. We also provide several criteria for when this τ-boundary is maximal. Our technique can for instance be used to construct examples of finitely supported random walks on certain lamplighter groups and solvable Baumslag-Solitar groups, whose Poisson boundaries are prime, but not L^p-irreducible for any p ≥ 1, answering a conjecture of Bader-Muchnik in the negative. Furthermore, we provide the first example of a countable discrete group Γ and two spread-out probability measures τ1 and τ2 on Γ such that the boundary entropy spectrum of (Γ, τ1) is an interval, while the boundary entropy spectrum of (Γ, τ2) is a Cantor set.