Title:
Bounded cohomology and hyperbolic groups
Abstract:
It was stated by M. Gromov that, for any hyperbolic group G, the map
from bounded cohomology H^{n}_{b}(G,R) to
H^{n}(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 H ^{n}_{b}(G,V) -> H^{n}(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 H ^{2}_{b}(G,V) -> H^{2}(G,V) is surjective for
any bounded G-module V, then G is hyperbolic. This gives a
characterization of hyperbolic groups by bounded cohomology.
The papers are available via Internet at http://www.math.usouthal.edu/~mineyev/math/ |

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