Measure conjugacy invariants for actions of countable sofic groups

Bowen, Lewis

doi:10.1090/S0894-0347-09-00637-7

Measure conjugacy invariants for actions of countable sofic groups
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by Lewis Bowen

J. Amer. Math. Soc. 23 (2010), 217-245

DOI: https://doi.org/10.1090/S0894-0347-09-00637-7

Published electronically: April 29, 2009

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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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