  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.
