Ergodic equivalence relations, cohomology, and von Neumann algebras. I

Let $(X,\mathcal{B})$ be a standard Borel space, $R \subset X \times X$ an equivalence relation $\in \mathcal{B} \times \mathcal{B}$. Assume each equivalence class is countable. Theorem 1: $\exists$ a countable group G of Borel isomorphisms of $(X,\mathcal{B})$ so that $R = \{ (x,gx):g \in G\}$. 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 [1]-[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 $\alpha ,\beta$ be rationally independent irrationals on the circle $\mathbb{T}$, and f Borel: $\mathbb{T} \to \mathbb{T}$. Then $\exists$ Borel $g,h:\mathbb{T} \to \mathbb{T}$ with $f(x) = (g(ax)/g(x))(h(\beta x)/h(x))$ 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].