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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
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