Random walks on Baumslag-Solitar groups

TL;DR

This paper studies the long-term behavior of random walks on Baumslag-Solitar groups by identifying their Poisson-Furstenberg boundary using geometric and probabilistic tools.

Contribution

It provides a detailed description of the Poisson-Furstenberg boundary for non-amenable Baumslag-Solitar groups, linking it to the space of ends of the Bass-Serre tree and the real line.

Findings

Identified the Poisson-Furstenberg boundary for BS(p,q) groups.

Connected the boundary to the space of ends of the Bass-Serre tree.

Used Kaimanovich's strip criterion to characterize the boundary.

Abstract

We consider random walks on non-amenable Baumslag-Solitar groups BS(p,q) and describe their Poisson-Furstenberg boundary. The latter is a probabilistic model for the long-time behaviour of the random walk. In our situation, we identify it in terms of the space of ends of the Bass-Serre tree and the real line using Kaimanovich's strip criterion.