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Measure conjugacy invariants for actions of countable sofic groups


Author: Lewis Bowen
Journal: J. Amer. Math. Soc. 23 (2010), 217-245
MSC (2000): Primary 37A35
DOI: https://doi.org/10.1090/S0894-0347-09-00637-7
Published electronically: April 29, 2009
MathSciNet review: 2552252
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Abstract: Sofic groups were defined implicitly by Gromov and explicitly by Weiss. All residually finite groups (and hence all linear groups) are sofic. The purpose of this paper is to introduce, for every countable sofic group $ G$, a family of measure-conjugacy invariants for measure-preserving $ G$-actions on probability spaces. These invariants generalize Kolmogorov-Sinai entropy for actions of amenable groups. They are computed exactly for Bernoulli shifts over $ G$, leading to a complete classification of Bernoulli systems up to measure-conjugacy for many groups, including all countable linear groups. Recent rigidity results of Y. Kida and S. Popa are utilized to classify Bernoulli shifts over mapping class groups and property (T) groups up to orbit equivalence and von Neumann equivalence, respectively.


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Additional Information

Lewis Bowen
Affiliation: Department of Mathematics, University of Hawaii, 2565 McCarthy Mall, Keller 409, Honolulu, HI 96822
Email: lpbowen@math.hawaii.edu

DOI: https://doi.org/10.1090/S0894-0347-09-00637-7
Keywords: Entropy, Ornstein's isomorphism theorem, Bernoulli shifts, measure conjugacy, orbit equivalence, von Neumann equivalence, sofic groups, group measure space construction
Received by editor(s): August 20, 2008
Published electronically: April 29, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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