Furstenberg entropy of intersectional invariant random subgroups

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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.

Original languageEnglish
Pages (from-to)2239-2265
Number of pages27
JournalCompositio Mathematica
Volume154
Issue number10
DOIs
StatePublished - 1 Oct 2018

Keywords

  • Furstenberg entropy
  • invariant random subgroup
  • stationary actions

ASJC Scopus subject areas

  • Algebra and Number Theory

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