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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Modular actions and amenable representations
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by Inessa Epstein and Todor Tsankov PDF
Trans. Amer. Math. Soc. 362 (2010), 603-621 Request permission

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
  • 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