Relative Bounded Cohomology for Groupoids

TL;DR

This paper extends bounded cohomology theory to groupoids, establishing isometric isomorphisms with topological and amenable groupoid pairs, generalizing previous results and providing a new algebraic framework.

Contribution

It introduces bounded cohomology for groupoids, develops homological algebra tools, and generalizes key results to this broader setting.

Findings

Bounded cohomology of topological space pairs is isometrically isomorphic to that of their fundamental groupoids.

Bounded cohomology relative to an amenable groupoid is isometrically isomorphic to that of the larger groupoid.

The paper generalizes previous results of Ivanov, Frigerio, and Pagliantini to groupoids.

Abstract

We introduce bounded cohomology for (pairs of) groupoids and develop homological algebra to deal with it. We generalise results of Ivanov, Frigerio and Pagliantini to this setting and show that (under topological conditions) the bounded cohomology of a pair of topological spaces is isometrically isomorphic to the bounded cohomology of the pair of fundamental groupoids. Furthermore, we prove that bounded cohomology relative to an amenable groupoid is isometrically isomorphic to the bounded cohomology of the ambient groupoid.