# Random walks on dense subgroups of locally compact groups

Michael Björklund, Yair Hartman, Hanna Oppelmayer

Research output: Working paper/PreprintPreprint

## Abstract

Let $\Gamma$ be a countable discrete group, $H$ a lcsc totally disconnected group and $\rho : \Gamma \rightarrow H$ 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 $p \geq 1$, 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.
Original language English GB Published - 2020

### Publication series

Name Arxiv preprint

## Keywords

• Mathematics - Dynamical Systems
• Mathematics - Group Theory
• Mathematics - Probability

## Fingerprint

Dive into the research topics of 'Random walks on dense subgroups of locally compact groups'. Together they form a unique fingerprint.