On vanishing and Hausdorffness of asymptotic cohomology

Abstract

Asymptotic cohomology was recently introduced to study stability problems in group theory. This paper studies vanishing and Hausdorffness questions around asymptotic cohomology. Our first main result is that, in a fixed degree and for a fixed asymptotic coefficient module, vanishing of asymptotic cohomology is equivalent to vanishing of reduced asymptotic cohomology. In the special case of diagonal coefficient modules, we also obtain a full characterisation of vanishing in terms of bounded cohomology. Our second main result is that asymptotic cohomology vanishes in degree 1 for uniformly convex asymptotic Banach modules. From this, we can deduce Hausdorffness of degree-2 bounded cohomology under the same uniform convexity assumption.