Low-dimensional bounded cohomology and extensions of groups

TL;DR

This paper explores the relationship between bounded cohomology and group extensions, aiming to provide a new interpretative framework for bounded cohomology using quasihomomorphisms.

Contribution

It introduces a novel interpretation of bounded cohomology in terms of group extensions, extending classical results to the bounded setting with quasihomomorphisms.

Findings

Established a connection between bounded cohomology and group extensions.

Provided a framework to compute bounded cohomology using quasihomomorphisms.

Enhanced understanding of the structure of bounded cohomology in geometric group theory.

Abstract

Bounded cohomology of groups was first studied by Gromov in 1982. Since then it has sparked much research in Geometric Group Theory. However, it is notoriously hard to explicitly compute bounded cohomology, even for most basic `non-positively curved' groups. On the other hand, there is a well-known interpretation of ordinary group cohomology in dimension 2 and 3 in terms of group extensions. The aim of this paper is to make this interpretation available for bounded group cohomology. This will involve quasihomomorphisms as defined and studied by Fujiwara and Kapovich.