The boundary action of a sofic random subgroup of the free group

TL;DR

This paper proves that the boundary action of a sofic random subgroup of a free group is conservative, addressing a key open question and exploring related properties like cogrowth and limit sets.

Contribution

It establishes the conservativity of boundary actions for sofic random subgroups and investigates their asymptotic properties using Schreier graph analysis.

Findings

Boundary action of a sofic random subgroup is conservative

Analysis of cogrowth and limit sets of sofic random subgroups

Use of Schreier graph asymptotics to study subgroup properties

Abstract

We prove that the boundary action of a sofic random subgroup of a finitely generated free group is conservative. This addresses a question asked by Grigorchuk, Kaimanovich, and Nagnibeda, who studied the boundary actions of individual subgroups of the free group. Following their work, we also investigate the cogrowth and various limit sets associated to sofic random subgroups. We make heavy use of the correspondence between subgroups and their Schreier graphs, and central to our approach is an investigation of the asymptotic density of a given set inside of large neighborhoods of the root of a sofic random Schreier graph.