ℓ∞-Cohomology: Amenability, relative hyperbolicity, isoperimetric inequalities and undecidability

We revisit Gersten’s Formula: see text-cohomology of groups and spaces, removing the finiteness assumptions required by the original definition while retaining its geometric nature. Mirroring the corresponding results in bounded cohomology, we provide a characterization of amenable groups using Formula: see text-cohomology, and generalize Mineyev’s characterization of hyperbolic groups via Formula: see text-cohomology to the relative setting. We then describe how Formula: see text-cohomology is related to isoperimetric inequalities. We also consider some algorithmic problems concerning Formula: see text-cohomology and show that they are undecidable. In Appendix A, we prove a version of the de Rham’s theorem in the context of Formula: see text-cohomology.