Injective hulls of certain discrete metric spaces and groups

Details Injective hulls of certain discrete metric spaces and groups Injective hulls of certain discrete metric spaces and groups

2011-07-29

arxiv.org/abs/1107.5971v1

Mathematics

Group Theory

Scientific paper

Injective metric spaces, or absolute 1-Lipschitz retracts, share a number of properties with CAT(0) spaces. Isbell showed that every metric space X has an injective hull E(X). We prove 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 l^n_\infty, for each n. This applies to a class of finitely generated groups \Gamma, including word hyperbolic and abelian groups, among others. Then \Gamma\ acts properly on E(\Gamma) by cellular isometries, and the first barycentric subdivision of E(\Gamma) is a model for the classifying space \underbar{E}\Gamma\ for proper actions. If \Gamma\ is word hyperbolic, E(\Gamma) is finite dimensional and the action is cocompact; the injective hull thus provides an alternative to the Rips complex, with some extra features.

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