Poisson boundary of group extensions

TL;DR

This paper characterizes when finitely generated linear groups have non-trivial Poisson-Furstenberg boundaries, resolving the Stability Problem for these groups and providing new criteria for boundary triviality and non-triviality.

Contribution

It offers a characterization of non-trivial boundaries for finitely generated linear groups in characteristic p and establishes the Stability Problem's positive resolution in this class.

Findings

Characterization of non-trivial boundary for linear groups in characteristic p

Positive answer to the Stability Problem for these groups

Sufficient condition for boundary triviality in characteristic 0

Abstract

Given a finitely generated group, the well-known Stability Problem asks whether the non-triviality of the Poisson-Furstenberg boundary (which is equivalent to the existence of non-constant bounded harmonic functions) depends on the choice of simple random walk on the group. This question was far from being understood even in the class of linear groups. Given an amenable group, e.g. a solvable group, there is no known characterisation, even a conjectural one, of when it admits a simple random walk with non-trivial boundary. We provide a characterisation of groups with non-trivial boundary for finitely generated linear groups of characteristic $p$. We prove in particular that the Stability Problem has a positive answer in this class of groups. For linear groups of characteristic $0$, we prove a sufficient condition for the triviality of the boundary which does not depend on the choice of a simple random walk. We conjecture that our sufficient condition is also necessary. Our arguments are based on a new comparison criterion for group extensions, on new $\Delta$-restriction entropy estimates and a criterion for boundary non-triviality, and on a new "cautiousness" criterion for triviality of the boundary.