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Published 2009 | Published
Journal Article Open

Subequivalence relations and positive-definite functions

Abstract

We study a positive-definite function associated with a countable, measure-preserving equivalence relation, which can be used to measure quantitatively the proximity of subequivalence relations. Combined with a co-inducing construction introduced by Epstein and earlier work of Ioana, this can be used to construct many mixing actions of countable groups and establish the non-classifiability, in a strong sense, of orbit equivalence of actions of non-amenable groups. We also discuss connections with percolation on Cayley graphs and the theory of costs.

Additional Information

© 2009 EMS Publishing House. The research of A.S.K. and T.T. was partially supported by NSF Grant DMS-0455285. We would like to thank I. Epstein for allowing us to include here our joint results. We would also like to thank R. Lyons, S. Popa and Y. Shalom for many valuable comments.

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