INJECTIVE HULLS OF CERTAIN DISCRETE METRIC SPACES AND GROUPS

Abstract

Injective metric spaces, or absolute 1-Lipschitz retracts, share a number of properties with CAT(0) spaces. In the '60s Isbell showed that every metric space X has an injective hull E (X). Here it is proved that if X is the vertex set of a connected locally finite graph with a uniform stability property of intervals, then E (X) is a locally finite polyhedral complex with finitely many isometry types of n-cells, isometric to polytopes in [Formula: see text], for each n. This applies to a class of finitely generated groups Γ, including all word hyperbolic groups and abelian groups, among others. Then Γ acts properly on E(Γ) by cellular isometries, and the first barycentric subdivision of E(Γ) is a model for the classifying space [Formula: see text] for proper actions. If Γ is hyperbolic, E(Γ) is finite dimensional and the action is cocompact. In particular, every hyperbolic group acts properly and cocompactly on a space of non-positive curvature in a weak (but non-coarse) sense.