Hyperbolicity and bounded-valued cohomology

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 $FP_2(\mathbb Q)$ and for those satisfying a rational homological linear isoperimetric inequality, answering a question of Arora and Mart\'{i}nez-Pedroza.