Comparison of cocycles of measured equivalence relationsand lifting problems

Abstract

Let ${\cal R}$ be an ergodic discrete equivalence relation on a Lebesgue space and $\alpha$ its cocycle with values in a locally compact group $G$. We say that an automorphism of ${\cal R}$ is compatible with $\alpha$ if it preserves the cohomology class of $\alpha$. We introduce a quasi-order relation on the set of all cocycles of ${\cal R}$ by means of comparison of the corresponding groups of all automorphisms being compatible with them. We find simple necessary and sufficient conditions under which two cocycles of a hyperfinite measure-preserving equivalence relation with values in compact (possibly different) groups are connected by this relation. Next, given an ergodic subrelation ${\cal S}$ of ${\cal R}$, we investigate the problem of extending ${\cal S}$-cocycles up to ${\cal R}$-cocycles and improve the recent results of Gabriel–Lemańczyk–Schmidt. As an application we study the problem of lifting of automorphisms of ${\cal R}$ up to automorphisms of the skew product ${\cal R}\times_\alpha G$, provide new short proofs of some known results and answer several questions from [ALV, ALMN].