Let be a proper CAT(0)-space and let be a closed subgroup of the isometry group of . We show that if is non-elementary and contains a rank-one element then its second continuous bounded cohomology group with coefficients in the regular representation is non-trivial. As a consequence, up to passing to an open subgroup of finite index, either is a compact extension of a totally disconnected group or is a compact extension of a simple Lie group of rank one.
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Ursula Hamenstädt, Isometry groups of proper CAT(0)-spaces of rank one. Groups Geom. Dyn. 6 (2012), no. 3, pp. 579–618DOI 10.4171/GGD/166