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Furstenberg entropy of intersectional invariant random subgroups

Part of: Stochastic processes Ergodic theory

Published online by Cambridge University Press:  17 September 2018

Yair Hartman  and
Ariel Yadin
Yair Hartman
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA email hartman@math.northwestern.edu
Ariel Yadin
Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, 8410501 Be’er Sheva, Israel email yadina@bgu.ac.il
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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.

Keywords

stationary actionsFurstenberg entropyinvariant random subgroup

MSC classification

Primary: 37A40: Nonsingular (and infinite-measure preserving) transformations 37A15: General groups of measure-preserving transformations 37A50: Relations with probability theory and stochastic processes 60G10: Stationary processes

Information

Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 10 , October 2018 , pp. 2239 - 2265
Copyright
© The Authors 2018 

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Footnotes

Y. Hartman is supported by the Fulbright Post-Doctoral Scholar Program. A. Yadin is supported by the Israel Science Foundation (grant no. 1346/15). We would like to thank Lewis Bowen and Yair Glasner for helpful discussions. Thanks to an anonymous referee for comments and suggestions.

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