Fixed points and amenability in non-positive curvature

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.

Bibliographical: Math. Annalen,

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[Publications][Nicolas Monod]