Cohomological characterisation of hyperbolicity

TL;DR

This paper provides a cohomological criterion for hyperbolicity in geodesic metric spaces using second b5-cohomology, extending to relative and acylindrical hyperbolicity cases.

Contribution

It introduces a complete cohomological characterization of hyperbolicity and acylindrical hyperbolicity, extending previous geometric criteria to a cohomological framework.

Findings

Cohomological criterion for hyperbolicity via second b5-cohomology

Extension of the criterion to relative hyperbolic spaces with hyperbolic subgraphs

Cohomological characterization of acylindrical hyperbolicity

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.