Median quasimorphisms on CAT(0) cube complexes and their cup products

TL;DR

This paper extends vanishing results of cup products from Brooks quasimorphisms on free groups to median quasimorphisms on CAT(0) cube complexes, with implications for bounded cohomology of various groups.

Contribution

It introduces vanishing results for cup products of median quasimorphisms on CAT(0) cube complexes, generalizing previous work on free groups.

Findings

Vanishing of cup products for median quasimorphisms on groups acting on trees.

Vanishing results for right-angled Artin groups.

Potential applications in bounded cohomology theory.

Abstract

Cup products provide a natural approach to access higher bounded cohomology groups. We extend vanishing results on cup products of Brooks quasimorphisms of free groups to cup products of median quasimorphisms, i.e., Brooks-type quasimorphisms of group actions on CAT(0) cube complexes. In particular, we obtain such vanishing results for groups acting on trees and for right-angled Artin groups. Moreover, we outline potential applications of vanishing results for cup products in bounded cohomology.