Binate groups and separable-coefficient bounded acyclicity

Ascertain whether every binate group is X_sep-boundedly acyclic, i.e., whether for all separable dual Banach Γ-modules E the bounded cohomology H_b^k(Γ; E) vanishes for all k > 0; more generally, determine whether there exist groups that are boundedly acyclic for trivial real coefficients but fail to be X_sep-boundedly acyclic, and, if such groups exist, identify effective criteria to detect non-vanishing of bounded cohomology for some separable dual coefficients.

Background

Binate groups are known to be boundedly acyclic with trivial real coefficients, by Theorem 2.7 in the paper. The stronger notion of X_sep-bounded acyclicity asks for vanishing of bounded cohomology in all positive degrees for all separable dual Banach Γ-modules, a property satisfied by several displacement-based classes (e.g., groups with commuting cyclic conjugates).

Understanding whether the binate condition alone ensures X_sep-bounded acyclicity is central to clarifying the landscape of displacement techniques and their cohomological implications. The authors also pose a broader question about the existence of groups whose bounded cohomology vanishes for trivial coefficients but not for all separable dual coefficients, and how one might detect such failures of vanishing.