Cohomological characterisation of hyperbolicity

TL;DR

The paper provides a comprehensive cohomological characterisation of hyperbolicity for geodesic metric spaces by linking hyperbolicity to the vanishing of second $\ell^{\infty}$-cohomology under a finite homological isoperimetric condition, and extends these ideas to relative settings with uniformly hyperbolic subgraphs. It develops a robust $\ell^{\infty}$-cohomology theory on graphs, proves key extension lemmas for $1$-cocycles, and establishes excision and group-equivalences to translate between graph-level and group-level statements. The authors then apply these tools to give a cohomological criterion for hyperbolically embedded subgroups and, consequently, for acylindrical hyperbolicity, with a consistent framework for relative hyperbolicity via cusped spaces. Overall, the work unifies hyperbolicity, relative hyperbolicity, and acylindrical hyperbolicity under a cohomological vanishing paradigm, providing new avenues for quasi-isometry invariant analysis and subgroup embedding criteria in geometric group theory.

Abstract

For any geodesic metric space $X$, we give a complete cohomological characterisation of the hyperbolicity of $X$ in terms of vanishing of its second $\ell^{\infty}$-cohomology. We extend this result to the relative setting of $X$ with a collection of uniformly hyperbolic subgraphs. As an application, we give a cohomological characterisation of acylindrical hyperbolicity.

Figures (4)

• Figure 1: The triangle $\Delta$ and the polygon $H$.
• Figure 2: The filling $f$ of $h$.
• Figure 3: A graph $Z$ with two subgraphs $Y_1,Y_2$.
• Figure 4: Proof that $f$ is Lipschitz. All pairs of points that lie in the same element of $\mathcal{Y}$ turn out to be close to each other, due to projections being Lipschitz.

Theorems & Definitions (69)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Remark 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Definition 2.6