TY - JOUR

T1 - Random walks on dense subgroups of locally compact groups

AU - Björklund, Michael

AU - Hartman, Yair

AU - Oppelmayer, Hanna

PY - 2020

Y1 - 2020

N2 - 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.

AB - 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.

KW - math.DS

KW - math.GR

KW - math.PR

M3 - Article

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

ER -