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

We revisit Gersten's ℓ∞-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 ℓ∞-cohomology, and generalize Mineyev's characterization of hyperbolic groups via ℓ∞-cohomology to the relative setting. We then describe how ℓ∞-cohomology is related to isoperimetric inequalities. We also consider some algorithmic problems concerning ℓ∞-cohomology and show that they are undecidable. In Appendix A, we prove a version of the de Rham's theorem in the context of ℓ∞-cohomology.