Comparison of cocycles of measured equivalence relationsand lifting problems

Abstract

Let ${\cal R}$ be an ergodic discreteequivalence relation on a Lebesgue space and$\alpha$ its cocycle with values in a locally compact group $G$. Wesay thatan automorphism of ${\cal R}$ is compatible with $\alpha$ if itpreserves thecohomology class of $\alpha$. We introduce aquasi-order relation on the set ofall cocycles of ${\cal R}$ by means ofcomparison of the corresponding groups of all automorphisms beingcompatiblewith them. We find simple necessary and sufficient conditions underwhichtwo cocycles of a hyperfinite measure-preserving equivalence relationwithvalues in compact (possibly different) groups are connected by thisrelation. Next, given an ergodic subrelation ${\cal S}$ of ${\cal R}$, weinvestigate the problem of extending ${\cal S}$-cocyclesup to ${\cal R}$-cocyclesand improve the recent results of Gabriel–Lemańczyk–Schmidt. As anapplication we study the problem of lifting of automorphisms of${\cal R}$up to automorphisms of the skew product ${\cal R}\times_\alpha G$,providenew short proofs of some known results and answerseveral questions from [ALV, ALMN].