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Furstenberg entropy of intersectional invariant random subgroups

Published online by Cambridge University Press:  17 September 2018

Yair Hartman
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730, USA email [email protected]
Ariel Yadin
Affiliation:
Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, 8410501 Be’er Sheva, Israel email [email protected]

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.

Type
Research Article
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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