Generalizing Aligned Cochains to Equivariant Bounded Cohomology on CAT(0) Cube Complexes

Develop a generalization of the aligned cochain complex to define a suitable subcomplex of equivariant bounded cochains for group actions on CAT(0) cube complexes, enabling aligned-cochain-based arguments in that setting.

Background

Aligned cochains play a key role in proofs of vanishing results in the bounded cohomology of free groups, as used by Amontova and Bucher. However, for actions on CAT(0) cube complexes, existing work avoids aligned cochains due to the lack of an appropriate generalization.

The present paper circumvents this by introducing weight quasimorphisms to give unified vanishing proofs, but the methodological gap regarding aligned cochains in the equivariant bounded setting remains explicitly noted.