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

Abstract

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.