Hyperbolicity and bounded-valued cohomology

Abstract We generalise a theorem of Gersten on surjectivity of the restriction map in ^ ℓ ∞ -cohomology of groups. This leads to applications on subgroups of hyperbolic groups, quasi-isometric distinction of finitely generated groups and ^ ℓ ∞ -cohomology calculations for some well-known classes of groups. Along the way, we obtain hyperbolicity criteria for groups of type FP₂ ({ {Q}}) F P 2 (Q) and for those satisfying a rational homological linear isoperimetric inequality, answering a question of Arora and Martínez-Pedroza.