Fixed points and amenability in non-positive curvature (Q2392795)

The authors give an algebraic characterisation of amenable groups of isometries of a proper cocompact CAT\((0)\) space \(X\). For amenable discrete subgroups, they obtain a narrower description, implying Q-linearity in the torsion free case. Generalising the case of linear algebraic groups to Is\((X)\), they establish Levi decompositions for stabilisers of points at infinity of \(X\). A geometric counterpart of this sheds light on the refined bordification of \(X\) and leads to a converse to the Adam-Ballman theorem. They prove that unimodular cocompact groups cannot fix any point at infinity except in the Euclidean factor. The paper contains various fixed point results as illustrations.