Random walks on dense subgroups of locally compact groups

Abstract

Let $\Gamma$ be a countable discrete group, $H$ a lcsc totally disconnected group and $\rho :\Gamma \to H$ a homomorphism with dense image. We develop a general and explicit technique which provides, for every compact open subgroup $L<H$ and bi-$L$-invariant probability measure $\theta$ on $H$, a Furstenberg discretization $\tau$ of $\theta$ such that the Poisson boundary of $(H,\theta )$ is a $\tau$-boundary. Among other things, this technique allows us 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 $p\ge 1$, answering a conjecture of Bader-Muchnik in the negative. Furthermore, we give an 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.

40 pages, 0 figures. Comments are welcome!