Hyperbolicity and bounded-valued cohomology

Abstract

We generalise a theorem of Gersten on surjectivity of the restriction map in ${\ell }^{\infty }$-cohomology of groups. This leads to applications on subgroups of hyperbolic groups, quasi-isometric distinction of finitely generated groups and ${\ell }^{\infty }$-cohomology calculations for some well-known classes of groups. Along the way, we obtain hyperbolicity criteria for groups of type $F{P}_2(\mathbb{Q})$ and for those satisfying a rational homological linear isoperimetric inequality, answering a question of Arora and Martínez-Pedroza.

24 pages. Final version, to appear in Math. Ann