TY - UNPB

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 - Mathematics - Dynamical Systems

KW - Mathematics - Group Theory

KW - Mathematics - Probability

M3 - פרסום מוקדם

T3 - Arxiv preprint

BT - Random walks on dense subgroups of locally compact groups

ER -