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Isometry groups of proper hyperbolic spaces. (English) Zbl 1273.53037
Summary: Let $$X$$ be a proper hyperbolic geodesic metric space and let $$G$$ be a closed subgroup of the isometry group $$\text{Iso}(X)$$ of $$X$$. We show that if $$G$$ is not elementary then for every $$p\in (1,\infty)$$ the second continuous bounded cohomology group $$H^2_{cb}(G,L^p(G))$$ does not vanish. As an application, we derive some structure results for closed subgroups of $$\text{Iso}(X)$$.

##### MSC:
 53C24 Rigidity results 20F67 Hyperbolic groups and nonpositively curved groups 20J06 Cohomology of groups
##### Keywords:
hyperbolic spaces; isometry groups; bounded cohomology
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