Growth conditions for freeness of the Furstenberg boundary action

TL;DR

This paper establishes conditions based on growth and decay functions that ensure the freeness of group actions on the Furstenberg boundary, and describes stationary states on the group's reduced C*-algebra.

Contribution

It provides new criteria linking growth conditions to the freeness of boundary actions and characterizes stationary states on reduced C*-algebras.

Findings

Freeness of boundary action is guaranteed under specific growth/decay conditions.

Support of stationary states on the reduced C*-algebra is explicitly described.

New connections between group growth properties and boundary dynamics.

Abstract

Given a finitely generated group $\Gamma$ and $g\in\Gamma$, we prove sufficient conditions in terms of various growth/decay functions for freeness of the action of $g$ on the Furstenberg boundary of $\Gamma$. In this context, we also give a description of the support of stationary states on the reduced C*-algebra of $\Gamma$.