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Modular actions and amenable representations

Epstein, Inessa and Tsankov, Todor (2010) Modular actions and amenable representations. Transactions of the American Mathematical Society, 362 (2). pp. 603-621. ISSN 0002-9947.

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Consider a measure-preserving action Γ ↷ (X, μ) of a countable group Γ and a measurable cocycle α: X × Γ → Aut(Y) with countable image, where (X, μ) is a standard Lebesgue space and (Y, ν) is any probability space. We prove that if the Koopman representation associated to the action Γ ↷ X is non-amenable, then there does not exist a countable-to-one Borel homomorphism from the orbit equivalence relation of the skew product action Γ ↷^α X × Y to the orbit equivalence relation of any modular action (i.e., an inverse limit of actions on countable sets or, equivalently, an action on the boundary of a countably-splitting tree), generalizing previous results of Hjorth and Kechris. As an application, for certain groups, we connect antimodularity to mixing conditions. We also show that any countable, non-amenable, residually finite group induces at least three mutually orbit inequivalent free, measure-preserving, ergodic actions as well as two non-Borel bireducible ones.

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Additional Information:© 2009 American Mathematical Society. Received by editor(s): March 22, 2007; received by editor(s) in revised form: April 12, 2007; posted: September 14, 2009. The first author’s research was partially supported by NSF grant 443948-HJ-21632. The second author’s research was partially supported by NSF grant and DMS-0455285. The authors would like to thank their respective advisors G. Hjorth and A. S. Kechris for encouragement, support, and valuable discussions on the topic of this paper. The authors are also grateful to the anonymous referee for suggesting a simplified proof of Lemma 5.1.
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Subject Keywords:Modular actions; amenable representations; orbit equivalence; Borel reducibility.
Issue or Number:2
Classification Code:Primary 37A20; Secondary 22D10.
Record Number:CaltechAUTHORS:20100202-111929477
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:17371
Deposited By: Jason Perez
Deposited On:10 Feb 2010 23:34
Last Modified:03 Oct 2019 01:27

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