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.
Cite this article
Adrian Ioana, Alexander S. Kechris, Todor Tsankov, Subequivalence relations and positive-definite functions. Groups Geom. Dyn. 3 (2009), no. 4 pp. 579–625