Bounded cohomology and isometry groups of hyperbolic spaces

  • Ursula Hamenstädt

    Universität Bonn, Germany


Let XX be an arbitrary hyperbolic geodesic metric space and let Γ\Gamma be a countable subgroup of the isometry group Iso(X){\rm Iso}(X) of XX. We show that if Γ\Gamma is non-elementary and weakly acylindrical (this is a weak properness condition) then the second bounded cohomology groups Hb2(Γ,R)H_b^2(\Gamma,\mathbb{R}), Hb2(Γ,p(Γ))H_b^2(\Gamma,\ell^p(\Gamma)) (1<p<)(1< p <\infty) 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.

Cite this article

Ursula Hamenstädt, Bounded cohomology and isometry groups of hyperbolic spaces. J. Eur. Math. Soc. 10 (2008), no. 2, pp. 315–349

DOI 10.4171/JEMS/112