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.