Bounded cohomology and hyperbolic groups
Abstract: It was stated by M. Gromov that, for any hyperbolic group G, the map from bounded cohomology Hnb(G,R) to Hn(G,R) induced by inclusion is surjective for n\geq 2. This surjectivity statement was used by A. Connes and H. Moscovici to show the Novikov conjecture for hyperbolic groups.
We introduce a homological analog of straightening simplices, which works for any hyperbolic group. This implies that the map Hnb(G,V) -> Hn(G,V) is surjective for n\geq 2 when V is
1. a bounded G-module or
2. a finitely generated abelian group.
Conversely, if G is finitely presentable and the map H2b(G,V) -> H2(G,V) is surjective for any bounded G-module V, then G is hyperbolic. This gives a characterization of hyperbolic groups by bounded cohomology.
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