Vanishing of Cup Products with Arbitrary Degree-2 Classes in Free Group Bounded Cohomology

Establish whether, for every non-abelian free group F and all k > 0, the cup products Hb^k(F; R) × Hb^2(F; R) → Hb^{k+2}(F; R) and Hb^2(F; R) × Hb^k(F; R) → Hb^{k+2}(F; R) are always trivial when the degree-2 factor is an arbitrary bounded cohomology class.

Background

Because every degree-2 bounded cohomology class of a free group is represented by the coboundary of a quasimorphism, one strategy to probe higher degrees is to take cup products with degree-2 classes.

Several partial results establish vanishing in special cases (e.g., Brooks, Δ-decomposable, and median quasimorphisms), but a general vanishing statement for cup products with arbitrary degree-2 classes remains unresolved.