Cohomological Characterisation of Hyperbolicity

For any geodesic metric space X, we give a complete cohomological characterisation of the hyperbolicity of X in terms of vanishing of its second ∞-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.