Boundaries of dense subgroups of totally disconnected groups

Abstract

Let Γ \Gamma be a countable discrete group and let H H be a lcsc totally disconnected group, L L a compact open subgroup of H H , and ρ : Γ → H \rho : \Gamma \rightarrow H a homomorphism with dense image. In this paper we construct, for every bi- L L -invariant probability measure θ \theta on H H , an explicit Furstenberg discretization τ \tau of θ \theta such that the Poisson boundary ( B θ , ν θ ) (B_\theta ,\nu _\theta ) of ( H , θ ) (H,\theta ) is a τ \tau -boundary, where Γ \Gamma acts on B θ B_\theta via the homomorphism ρ \rho . We also provide several criteria for when this τ \tau -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 L^p -irreducible for any p ≥ 1 p \geq 1 , answering a conjecture of Bader-Muchnik in the negative. Furthermore, we provide the first example of a countable discrete group Γ \Gamma and two spread-out probability measures τ 1 \tau _1 and τ 2 \tau _2 on Γ \Gamma such that the boundary entropy spectrum of ( Γ , τ 1 ) (\Gamma ,\tau _1) is an interval, while the boundary entropy spectrum of ( Γ , τ 2 ) (\Gamma ,\tau _2) is a Cantor set.