An ergodic theorem for the quasi-regular representation of the free group

TL;DR

This paper extends ergodic theorems and irreducibility results for group actions on boundaries to the free group, even with potential arithmetic spectra, broadening the scope of previous geometric group theory results.

Contribution

It generalizes ergodic theorems and irreducibility of quasi-regular representations to free groups, regardless of spectrum arithmeticity, filling a gap in the theory for non-manifold groups.

Findings

Proves ergodic theorems for free groups

Establishes irreducibility of associated unitary representations

Applies to free groups with arithmetic spectra

Abstract

In \cite{BAMU}, an ergodic theorem \`a la Birkhoff-von Neumann for the action of the fundamental group of a compact negatively curved manifold on the boundary of its universal cover is proved. A quick corollary is the irreducibility of the associated unitary representation. These results are generalized \cite{BOYER} to the context of convex cocompact groups of isometries of a CAT(-1) space, using Theorem 4.1.1 of \cite{ROBLI}, with the hypothesis of non arithmeticity of the spectrum. We prove all the analog results in the case of the free group $\mathbb{F}_r$ of rank $r$ even if $\mathbb{F}_r$ is not the fundamental group of a closed manifold, and may have an arithmetic spectrum.