Consider a proper cocompact CAT(0) space X. We give a complete algebraic characterisation of amenable groups of isometries of X. For amenable discrete subgroups, an even narrower description is derived, implying Q-linearity in the torsion-free case. We establish Levi decompositions for stabilisers of points at infinity of X, generalising the case of linear algebraic groups to Isom(X). A geometric counterpart of this sheds light on the refined bordification of X (à la Karpelevich) and leads to a converse to the Adams–Ballmann theorem. It is further deduced that unimodular cocompact groups cannot fix any point at infinity except in the Euclidean factor; this fact is needed for the study of CAT(0) lattices. Various fixed point results are derived as illustrations.
Donna Testerman, Martin W. Liebeck
Romain Christophe Rémy Fleury, Haoye Qin, Aleksi Antoine Bossart, Zhechen Zhang