# Furstenberg Entropy of Intersectional Invariant Random Subgroups

## TL;DR

This paper demonstrates that for free groups and certain other groups, any entropy value can be realized through ergodic stationary actions constructed via invariant random subgroups, expanding the understanding of Furstenberg entropy realization.

## Contribution

It introduces a general framework for constructing a continuum of ergodic IRSs under algebraic conditions, enabling full entropy realization for specific groups.

## Key findings

- Any entropy value can be realized for free groups via ergodic stationary actions.
- Constructs a continuum of ergodic IRSs for certain groups, leading to a continuum of entropy values.
- Provides geometric and probabilistic methods to analyze entropy through Schreier graphs.

## Abstract

We study the Furstenberg-entropy realization problem for stationary actions. It is shown that for finitely supported probability measures on free groups, any a-priori possible entropy value can be realized as the entropy of an ergodic stationary action. This generalizes results of Bowen. The stationary actions we construct arise via invariant random subgroups (IRSs), based on ideas of Bowen and Kaimanovich. We provide a general framework for constructing a continuum of ergodic IRSs for a discrete group under some algebraic conditions, which gives a continuum of entropy values. Our tools apply for example, for certain extensions of the group of finitely supported permutations and lamplighter groups, hence establishing full realization results for these groups. For the free group, we construct the IRSs via a geometric construction of subgroups, by describing their Schreier graphs. The analysis of the entropy of these spaces is obtained by studying the random walk on the appropriate Schreier graphs.