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Extensive amenability and an application to interval exchanges

Published online by Cambridge University Press:  29 July 2016

KATE JUSCHENKO ,
NICOLÁS MATTE BON ,
NICOLAS MONOD  and
MIKAEL DE LA SALLE
KATE JUSCHENKO
Affiliation:
Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208, USA email kate.juschenko@gmail.com
NICOLÁS MATTE BON
Affiliation:
Université Paris-Sud & Ecole Normale Supérieure, DMA, 45 rue d’Ulm, 75230 Paris Cedex 05, France email nicolas.matte.bon@ens.fr
NICOLAS MONOD
Affiliation:
EPFL, 1015 Lausanne, Switzerland email nicolas.monod@epfl.ch
MIKAEL DE LA SALLE
Affiliation:
UMPA UMR CNRS 5669 - ENS de Lyon, 46 allée d’Italie, 69364, Lyon Cedex 07, France email mikael.de.la.salle@ens-lyon.fr
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Abstract

Extensive amenability is a property of group actions which has recently been used as a tool to prove amenability of groups. We study this property and prove that it is preserved under a very general construction of semidirect products. As an application, we establish the amenability of all subgroups of the group $\text{IET}$ of interval exchange transformations that have angular components of rational rank less than or equal to two. In addition, we obtain a reformulation of extensive amenability in terms of inverted orbits and use it to present a purely probabilistic proof that recurrent actions are extensively amenable. Finally, we study the triviality of the Poisson boundary for random walks on $\text{IET}$ and show that there are subgroups $G<\text{IET}$ admitting no finitely supported measure with trivial boundary.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 38 , Issue 1 , February 2018 , pp. 195 - 219
Copyright
© Cambridge University Press, 2016 

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