Random walks on dense subgroups of locally compact groups

2020/06/28 by Michael Björklund, Björklund, Michael, Yair Hartman +3 · 1 citation
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR) #math.DS #math.GR #math.PR

paper · pdf · doi:10.48550/arxiv.2006.15705

40 pages, 0 figures. Comments are welcome!

arxiv created 2020/06/28 · arxiv updated 2020/06/30

Abstract

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 L^p-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.

Cited by

• Quotients of Poisson boundaries, entropy, and spectral gap

Related

• Shannon's theorem for locally compact groups
• The Poisson boundary of discrete subgroups of semisimple Lie groups without moment conditions
• Random walks on Baumslag-Solitar groups
• Poisson boundary of group extensions
• Boundary of the boundary for random walks on groups