Cohomological characterisation of hyperbolicity

2024/09/27 by Milizia, Francesco, Petrosyan, Nansen, Sisto, Alessandro +1
#20F65 (Primary) 20F67 #57M07 (Secondary) #Algebraic Topology (math.AT) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2409.18871

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 ℓ^∞-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.

Related

• ℓ^∞-cohomology: amenability, relative hyperbolicity, isoperimetric inequalities and undecidability
• Hyperbolicity and bounded-valued cohomology
• Bounded cohomology and isometry groups of hyperbolic spaces
• On hyperbolic cohomology classes
• The zero norm subspace of bounded cohomology of acylindrically\n hyperbolic groups