Hyperbolicity and bounded-valued cohomology

We generalise a theorem of Gersten on surjectivity of the restriction map in & ell; infinity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell <^>{\infty }$$\end{document} -cohomology of groups. This leads to applications on subgroups of hyperbolic groups, quasi-isometric distinction of finitely generated groups and & ell; infinity \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell <^>{\infty }$$\end{document} -cohomology calculations for some well-known classes of groups. Along the way, we obtain hyperbolicity criteria for groups of type F P 2 ( Q ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$FP_2({{\mathbb {Q}}})$$\end{document} and for those satisfying a rational homological linear isoperimetric inequality, answering a question of Arora and Mart & iacute;nez-Pedroza.