Groups, Geometry, and Dynamics


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Volume 6, Issue 3, 2012, pp. 579–618
DOI: 10.4171/GGD/166

Published online: 2012-08-16

Isometry groups of proper CAT(0)-spaces of rank one

Ursula Hamenstädt[1]

(1) Universität Bonn, Germany

Let $X$ be a proper CAT(0)-space and let $G$ be a closed subgroup of the isometry group $\mathrm{Iso}(X)$ of $X$. We show that if $G$ 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 $G$ is a compact extension of a totally disconnected group or $G$ is a compact extension of a simple Lie group of rank one.

Keywords: Bounded cohomology, isometry groups, CAT(0)-spaces, rigidity

Hamenstädt Ursula: Isometry groups of proper CAT(0)-spaces of rank one. Groups Geom. Dyn. 6 (2012), 579-618. doi: 10.4171/GGD/166