TY - UNPB

T1 - Random walks on dense subgroups of locally compact groups

AU - Björklund, Michael

AU - Hartman, Yair

AU - Oppelmayer, Hanna

PY - 2020/6/28

Y1 - 2020/6/28

N2 - Let Γ be a countable discrete group, H a lcsc totally disconnected group and ρ:Γ→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 θ on H, a Furstenberg discretization τ of θ such that the Poisson boundary of (H,θ) is a τ-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 Lp-irreducible for any p≥1, answering a conjecture of Bader-Muchnik in the negative. Furthermore, we give an example of a countable discrete group Γ and two spread-out probability measures τ1 and τ2 on Γ such that the boundary entropy spectrum of (Γ,τ1) is an interval, while the boundary entropy spectrum of (Γ,τ2) is a Cantor set.

AB - Let Γ be a countable discrete group, H a lcsc totally disconnected group and ρ:Γ→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 θ on H, a Furstenberg discretization τ of θ such that the Poisson boundary of (H,θ) is a τ-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 Lp-irreducible for any p≥1, answering a conjecture of Bader-Muchnik in the negative. Furthermore, we give an example of a countable discrete group Γ and two spread-out probability measures τ1 and τ2 on Γ such that the boundary entropy spectrum of (Γ,τ1) is an interval, while the boundary entropy spectrum of (Γ,τ2) is a Cantor set.

KW - Mathematics - Dynamical Systems

KW - Mathematics - Group Theory

KW - Mathematics - Probability

U2 - https://doi.org/10.48550/arXiv.2006.15705

DO - https://doi.org/10.48550/arXiv.2006.15705

M3 - Preprint

BT - Random walks on dense subgroups of locally compact groups

ER -