AF-equivalence relations and their cocycles

TL;DR

This paper explores AF-equivalence relations in topological settings, showing all cocycles are cohomologous to quasi-product cocycles and analyzing measures related to these cocycles.

Contribution

It introduces a topological version of AF-equivalence relations and characterizes cocycles as cohomologous to quasi-product cocycles, extending measured theory results.

Findings

Every cocycle is cohomologous to a quasi-product cocycle.

Characterization of measures admitting a cocycle as Radon-Nikodym derivative.

Extension of hyperfinite equivalence relation theory to topological context.

Abstract

After a review of some of the main results about hyperfinite equivalence relations and their cocycles in the measured setting, we give a definition of a topological AF-equivalence relation. We show that every cocycle is cohomologous to a quasi-product cocycle. We then study the problem of determining the quasi-invariant probability measures admitting a given cocycle as their Radon-Nikodym derivative.