Abstract
We study a positive-definite function associated with a countable, measure-preser- ving equivalence relation, which can be used to measure quantitatively the proximity of sube- quivalence 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. Mathematics Subject Classification (2000). 37A20, 03E15.
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