Median quasimorphisms on $${{\,\mathrm{{CAT}}\,}}(0)$$ cube complexes and their cup products

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 \({{\,\mathrm{{CAT}}\,}}(0)\) 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.