Fixed points and amenability in non-positive curvature

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 (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.

33 pages

References in corpus (6)

• Rank rigidity for CAT(0) cube complexes
• Isometry groups of non-positively curved spaces: structure theory
• A characterization of irreducible symmetric spaces and Euclidean buildings of higher rank by their asymptotic geometry
• Isometry groups of non-positively curved spaces: discrete subgroups
• Contracting automorphisms and L^p-cohomology in degree one
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• Rank rigidity for CAT(0) cube complexes
• The Median Class and Superrigidity of Actions on CAT(0) Cube Complexes
• Gelfand pairs and strong transitivity for Euclidean buildings
• Erratum and addenda to "Isometry groups of non-positively curved spaces: discrete subgroups"
• Lattices in amenable groups
• Groups acting on spaces of non-positive curvature
• Detecting product splittings of CAT(0) spaces
• A homogeneous ${\tilde{A}}_2$-building with a non-discrete automorphism group is Bruhat-Tits
• Representations of infinite dimension orthogonal groups of quadratic forms with finite index
• On ends of finite-volume noncompact manifolds of nonpositive curvature
• Strong transitivity, Moufang's condition and the Howe--Moore property