Modular actions and amenable representations
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- by Inessa Epstein and Todor Tsankov PDF
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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.References
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Additional Information
- Inessa Epstein
- Affiliation: Department of Mathematics, University of California, Mathematical Sciences Building 6363, Los Angeles, California 90095
- Email: iepstein@math.ucla.edu
- Todor Tsankov
- Affiliation: Department of Mathematics 253-37, California Institute of Technology, Pasadena, California 91125
- MR Author ID: 781832
- Email: todor@caltech.edu
- 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. - © Copyright 2009 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 362 (2010), 603-621
- MSC (2000): Primary 37A20; Secondary 22D10
- DOI: https://doi.org/10.1090/S0002-9947-09-04525-5
- MathSciNet review: 2551499