Fixed points and amenability in non-positive curvature

TL;DR

This paper provides a comprehensive algebraic characterization of amenable groups acting on proper cocompact CAT(0) spaces, extending known results and establishing new fixed point and structural theorems.

Contribution

It offers a complete algebraic description of amenable isometry groups of CAT(0) spaces and generalizes Levi decompositions to stabilizers at infinity.

Findings

Amenable groups are characterized algebraically in CAT(0) spaces.

Levi decompositions are established for stabilizers at infinity.

Unimodular cocompact groups cannot fix points at infinity outside Euclidean factors.

Abstract

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.