Cohomology groups invariant under continuous orbit equivalence

TL;DR

This paper demonstrates that certain cohomology groups related to group actions are invariant under continuous orbit equivalence, providing new tools for classifying topologically free actions.

Contribution

It establishes that variations of uniformly finite homology and bounded cohomology groups are invariants under continuous orbit equivalence for topologically free actions.

Findings

Cohomology groups are invariant under continuous orbit equivalence.

Topological amenability can be characterized via these invariants.

Results extend previous characterizations of group actions.

Abstract

By the work of Brodzki-Niblo-Nowak-Wright and Monod, topological amenability of a continuous group action can be characterized using uniformly finite homology groups or bounded cohomology groups associated to this action. We show that (certain variations of) these groups are invariants for topologically free actions under continuous orbit equivalence.