ℓ^∞-cohomology: amenability, relative hyperbolicity, isoperimetric inequalities and undecidability · arXivDesk
2107.09089Jul 19, 202136 pages. v3: added a characterization of relative hyperbolicity and some undecidability results. v4: changed the title, moved the section about differential forms into an appendix, made some changes to the introduction
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∞
Nearby in the stack
-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 an appendix, we prove a version of the de Rham's theorem in the context of