Ergodic Equivalence Relations, Cohomology, and Von Neumann Algebras. I

Let (X, ) be a standard Borel space, R c X X X an equivalence, relation e x S. Assume each equivalence class is countable. Theorem 1: 3a countable group G of Borel isomorphisms of (X, B) so that R -{ (x, gx): g e G). G is far from unique. However, notions like invariance and quasi-invariance and R-N derivatives of measures depend only on R, not the choice of G. We develop some of the ideas of Dye 1, 2 and Krieger l-5 in a fashion explicitly avoiding any choice of G; we also show the connection with virtual groups. A notion of "module over R" is defined, and we axiomatize and develop a cohomology theory for R with coefficients in such a module. Surprising application (contained in Theorem 7): let a, be rationally independent irrationals on the circle T, and/Borel: T-T. Then 3 Borel g, h: T->T with/ (x) - (g (ax) /g (x) ) (h (x) /h (x) ) a. e. The notion of "skew product action" is generalized to our context, and provides a setting for a generalization of the Krieger invariant for the R-N derivative of an ergodic transformation: we define, for a cocycle c on R with values in the group A, a subgroup of A depending only on the cohomology class of c, and in Theorem 8 identify this with another subgroup, the "normalized proper range" of c, defined in terms of the skew action. See also Schmidt 1. 1. Introduction. This is the first of a series of two papers which will provide the details of the results announced in Feldman-Moore 1. The first of these will be devoted more to a study of the equivalence relations and their cohomology, while the second will be devoted more to the application of these results and techniques to the study of von Neumann algebras. Throughout, X will be a standard Borel space with a-field. If G is some countable group of Borel automorphisms of X we introduce the orbit equivalence relation of this action, namely RG = { (x, y): 3g E G, y g x) c X X X. If p is a a-finite measure on X, we say that it is quasi-invariant if its null