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

TL;DR

This work extends $\ell^\infty$-cohomology to contexts without finiteness restrictions by defining $H_{(\infty)}^\bullet(G;V)=H^\bullet(G;\ell^\infty(G,V))$ and relating it to spaces via cochains bounded on orbits. It establishes a sharp amenability criterion via injectivity of comparison maps and generalizes Mineyev’s hyperbolicity characterization to relative settings, while linking $\ell^\infty$-cohomology to linear isoperimetric inequalities. The authors also develop relative $\ell^\infty$-cohomology for group pairs, prove strong vanishing results for relatively hyperbolic pairs, and demonstrate undecidability results for computing $\ell^\infty$-cohomology in all positive degrees. An appendix provides a de Rham-type theorem for $\ell^\infty$-cohomology, reinforcing the geometric and analytic robustness of the theory.

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.

Theorems & Definitions (61)

• Theorem 1.1
• Theorem 1.2: Mineyev2000
• Theorem 1.3
• Proposition 1.4: Gersten1992Wienhard2012Blank2015
• Theorem 1.5
• Theorem 1.6
• Definition 2.1
• Definition 2.2
• Remark 2.3
• Definition 2.4