Furstenberg entropy of intersectional invariant random subgroups
| Autor: | Yair Hartman, Ariel Yadin |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Pure mathematics
Algebra and Number Theory Discrete group Probability (math.PR) 010102 general mathematics Dynamical Systems (math.DS) Group Theory (math.GR) Random walk 01 natural sciences 010104 statistics & probability Free group FOS: Mathematics Ergodic theory Entropy (information theory) Mathematics - Dynamical Systems 0101 mathematics Algebraic number Mathematics - Group Theory Mathematics - Probability Probability measure Mathematics |
| Zdroj: | Compositio Mathematica. 154:2239-2265 |
| ISSN: | 1570-5846 0010-437X |
| DOI: | 10.1112/s0010437x18007261 |
| Popis: | 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. Comment: 31 pages |
| Databáze: | OpenAIRE |
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