Subequivalence relations and positive-definite functions

We study a positive-definite function associated to a measure-preservingequivalence relation on a standard probability space and use it to measurequantitatively the proximity of subequivalence relations. This is combined witha recent co-inducing construction of Epstein to produce new kinds of mixingactions of an arbitrary infinite discrete group and it is also used to showthat orbit equivalence of free, measure preserving, mixing actions ofnon-amenable groups is unclassifiable in a strong sense. Finally, in the caseof property (T) groups we discuss connections with invariant percolation onCayley graphs and the calculation of costs.