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 of a countable group and a measurable cocycle with countable image, where is a standard Lebesgue space and is any probability space. We prove that if the Koopman representation associated to the action is non-amenable, then there does not exist a countable-to-one Borel homomorphism from the orbit equivalence relation of the skew product action 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

Email:
todor@caltech.edu

DOI:
https://doi.org/10.1090/S0002-9947-09-04525-5

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