  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 Is(X). A geometric
counterpart of this sheds light on the refined bordification of X (\`a 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.
