Isometry groups of proper CAT.0/-spaces of rank one

Abstract

LetX be a properCAT.0/-space and letG be a closed subgroup of the isometry group 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. © European Mathematical Society.