${\ell }^{\infty }$-cohomology: amenability, relative hyperbolicity, isoperimetric inequalities and undecidability

Abstract

We revisit Gersten's ${\ell }^{\infty }$-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 ${\ell }^{\infty }$-cohomology, and generalize Mineyev's characterization of hyperbolic groups via ${\ell }^{\infty }$-cohomology to the relative setting. We then describe how ${\ell }^{\infty }$-cohomology is related to isoperimetric inequalities. We also consider some algorithmic problems concerning ${\ell }^{\infty }$-cohomology and show that they are undecidable. In an appendix, we prove a version of the de Rham's theorem in the context of ${\ell }^{\infty }$-cohomology.

arXiv comment36 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