Boundaries of dense subgroups of totally disconnected groups

Abstract:

Let $\Gamma$ be a countable discrete group and let $H$ be a lcsc totally disconnected group, $L$ a compact open subgroup of $H$, and $\rho : \Gamma \rightarrow H$ a homomorphism with dense image. In this paper we construct, for every bi-$L$-invariant probability measure $\theta$ on $H$, an explicit Furstenberg discretization $\tau$ of $\theta$ such that the Poisson boundary $(B_\theta ,\nu _\theta )$ of $(H,\theta )$ is a $\tau$-boundary, where $\Gamma$ acts on $B_\theta$ via the homomorphism $\rho$. We also provide several criteria for when this $\tau$-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 $L^p$-irreducible for any $p \geq 1$, answering a conjecture of Bader-Muchnik in the negative.

Furthermore, we provide the first example of a countable discrete group $\Gamma$ and two spread-out probability measures $\tau _1$ and $\tau _2$ on $\Gamma$ such that the boundary entropy spectrum of $(\Gamma ,\tau _1)$ is an interval, while the boundary entropy spectrum of $(\Gamma ,\tau _2)$ is a Cantor set.