Groups, Geometry, and Dynamics

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Volume 3, Issue 4, 2009, pp. 579–625
DOI: 10.4171/GGD/71

Published online: 2009-12-23

Subequivalence relations and positive-definite functions

Adrian Ioana[1], Alexander S. Kechris[2] and Todor Tsankov[3]

(1) University of California, San Diego, United States
(2) California Institute of Technology, Pasadena, United States
(3) Université Pierre et Marie Curie, Paris, France

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.

Keywords: Subequivalence relations, positive-definite functions, co-induced action, orbit equivalence, non-amenable groups, classification, percolation, property (T), cost

Ioana Adrian, Kechris Alexander, Tsankov Todor: Subequivalence relations and positive-definite functions. Groups Geom. Dyn. 3 (2009), 579-625. doi: 10.4171/GGD/71