Straightening and bounded cohomology of hyperbolic groups

Abstract.

It was stated by M. Gromov [Gr2] that, for any hyperbolic group G, the map from bounded cohomology $H^n_b(G,{\Bbb R})$ to $H^n(G,{\Bbb R})$ induced by inclusion is surjective for $n \ge 2$. We introduce a homological analogue of straightening simplices, which works for any hyperbolic group. This implies that the map $H^n_b(G,V) \to H^n(G,V)$ is surjective for $n \ge 2$ when V is any bounded ${\Bbb Q}G$-module and when V is any finitely generated abelian group.