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Bounded cohomology and isometry groups of hyperbolic spaces

Ursula Hamenstädt — 2008

Journal of the European Mathematical Society

Let X be an arbitrary hyperbolic geodesic metric space and let Γ be a countable subgroup of the isometry group Iso ( X ) of X . We show that if Γ is non-elementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups H b 2 ( Γ , ) , H b 2 ( Γ , p ( Γ ) ) ( 1 < p < ) are infinite dimensional. Our result holds for example for any subgroup of the mapping class group of a non-exceptional surface of finite type not containing a normal subgroup which virtually splits as a direct product....

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