Poisson boundary of groups acting on real trees

TL;DR

This paper provides a geometric characterization of the Poisson boundaries for specific group extensions, notably free-by-cyclic groups, using topological compactifications related to actions on real trees.

Contribution

It offers a new geometric description of Poisson boundaries for groups acting on real trees, including free-by-cyclic groups, expanding understanding of their boundary behavior.

Findings

Full description of Poisson boundaries for free-by-cyclic groups

Extension of Kaimanovich's topological compactification approach

Identification of groups with complex actions on real trees

Abstract

We give a geometric description of the Poisson boundaries of certain extensions of free and hyperbolic groups. In particular, we get a full description of the Poisson boundaries of free-by-cyclic groups. We rely upon the description of Poisson boundaries by means of a topological compactification as developed by Kaimanovich. All the groups studied here share the property of admitting a sufficiently complicated action on some real tree.