2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/219023Let G be a closed subgroup of the isometry group of a proper CAT(0)-space X. We show that if G is non-elementary and contains a rank-one element then its second 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.writing improved, some details added, 33 pagesGroup TheoryMetric Geometry20F67,20J06text