Hyperbolicity and bounded-valued cohomology (Q6624848)

Bounded-valued cohomology is a quasi-isometry invariant of finitely generated groups and it is an established tool to find lower bounds for the Dehn function of such groups. It is known due to \textit{S. M. Gersten} [J. Lond. Math. Soc., II. Ser. 54, No. 2, 261--283 (1996; Zbl 0861.20033)] that if \(G\) is a finitely presented group, then \(G\) is hyperbolic if and only if \(H^2_{(\infty)}(G;\ell^\infty(\mathbb{N},\mathbb{R})) = 0\). \N\NThis interesting paper under review generalizes a theorem of Gersten on surjectivity of the restriction map in bounded-valued cohomology of groups. This has applications to subgroups of hyperbolic groups, quasi-isometric distinction of finitely generated groups and calculations of bounded-valued cohomology for some well-known classes of groups. A group \(G\) is said to be of type \(FP_2(\mathbb{Q})\) if there is a projective \(\mathbb{Q}G\)-resolution \(\cdots \to P_2\to P_1\to P_0 \to \mathbb{Q}\) such that \(P_i\) is a finitely generated projective \(\mathbb{Q}G\)-module for each \(i\le 2\). The authors also obtain hyperbolicity criteria for groups of type \(FP_2(\mathbb{Q})\) and for those satisfying a rational homological linear isoperimetric inequality.