Ergodic properties of boundary actions and Nielsen--Schreier theory

TL;DR

This paper investigates the ergodic and conservative properties of subgroup actions on free group boundaries, linking these properties to Schreier graph growth and providing multiple equivalent descriptions of boundary decompositions.

Contribution

It introduces a geometric and combinatorial approach to analyze ergodic properties of subgroup actions, relating them to Schreier graph growth and Nielsen--Schreier generators.

Findings

Characterization of boundary ergodic properties via Schreier graph growth

Multiple equivalent descriptions of the Hopf decomposition

Construction of examples illustrating theoretical connections

Abstract

We study the basic ergodic properties (ergodicity and conservativity) of the action of an arbitrary subgroup $H$ of a free group $F$ on the boundary $\partial F$ with respect to the uniform measure. Our approach is geometrical and combinatorial, and it is based on choosing a system of Nielsen--Schreier generators in $H$ associated with a geodesic spanning tree in the Schreier graph $X=H\backslash F$. We give several (mod 0) equivalent descriptions of the Hopf decomposition of the boundary into the conservative and the dissipative parts. Further we relate conservativity and dissipativity of the action with the growth of the Schreier graph $X$ and of the subgroup $H$ ($\equiv$ cogrowth of $X$), respectively. We also construct numerous examples illustrating connections between various relevant notions.