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Modular actions and amenable representations

Authors: Inessa Epstein and Todor Tsankov
Journal: Trans. Amer. Math. Soc. 362 (2010), 603-621
MSC (2000): Primary 37A20; Secondary 22D10
Published electronically: September 14, 2009
MathSciNet review: 2551499
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Abstract: Consider a measure-preserving action $ \Gamma \curvearrowright (X, \mu)$ of a countable group $ \Gamma$ and a measurable cocycle $ \alpha \colon X \times \Gamma \to \mathrm{Aut}(Y)$ with countable image, where $ (X, \mu)$ is a standard Lebesgue space and $ (Y, \nu)$ is any probability space. We prove that if the Koopman representation associated to the action $ \Gamma \curvearrowright 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 $ \Gamma \curvearrowright^\alpha X \times 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

Inessa Epstein
Affiliation: Department of Mathematics, University of California, Mathematical Sciences Building 6363, Los Angeles, California 90095

Todor Tsankov
Affiliation: Department of Mathematics 253-37, California Institute of Technology, Pasadena, California 91125

Keywords: Modular actions, amenable representations, orbit equivalence, Borel reducibility
Received by editor(s): March 22, 2007
Received by editor(s) in revised form: April 12, 2007
Published electronically: September 14, 2009
Additional Notes: 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.
Article copyright: © Copyright 2009 American Mathematical Society

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