Poisson-Furstenberg boundary and growth of groups

TL;DR

This paper investigates the Poisson-Furstenberg boundary of random walks on permutational wreath products, providing conditions for non-trivial boundaries and constructing groups with exponential growth where all finitely supported random walks have trivial boundary.

Contribution

It introduces a new criterion for non-trivial boundaries in groups and constructs examples of exponential growth groups with trivial boundaries for all finitely supported walks.

Findings

Established a sufficient condition for non-trivial boundary existence.

Constructed exponential growth groups with trivial boundary for all finitely supported walks.

Provided a negative answer to a question by Kaimanovich and Vershik.

Abstract

We study the Poisson-Furstenberg boundary of random walks on permutational wreath products. We give a sufficient condition for a group to admit a symmetric measure of finite first moment with non-trivial boundary, and show that this criterion is useful to establish exponential word growth of groups. We construct groups of exponential growth such that all finitely supported (not necessarily symmetric, possibly degenerate) random walks on these groups have trivial boundary. This gives a negative answer to a question of Kaimanovich and Vershik.