Injective hulls of certain discrete metric spaces and groups

By U. Lang. 2011

132 citations recorded by the source corpus.

Injective metric spaces, or absolute 1-Lipschitz retracts, share a number of properties with CAT(0) spaces. In the 1960es, J. R. 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 l^n_\infty, for each n. This applies to a class of finitely generated groups G, including all word hyperbolic groups and abelian groups, among others. Then G acts properly on E(G) by cellular isometries, and the first barycentric subdivision of E(G) is a model for the classifying space \underbar{E}G for proper actions. If G is hyperbolic, E(G) 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.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/51d71bcff4434a971c535e9813f3e2d48c64bf36

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Weakly Modular Graphs and Nonpositive Curvature (2014, background): “g properties of corresponding groups. Every hyperbolic group acts geometrically on a nitely dimensional contractible ag simplicial complex associated to a Helly graph, that is on a Helly complex; cf. [Lan13]. Moreover, since thickenings of median graphs are Helly graphs, also CAT(0) cubical groups a”
Area Minimizing Discs in Metric Spaces (2015, background): “It moreover applies to complete metric spaces with a convex bicombing in the sense of [55] or [38] and, in particular, to all Banach spaces and all injective metric spaces.”
Intrinsic structure of minimal discs in metric spaces (2016, background): “Concerning the definition and properties of injective metric spaces and injective hulls we refer the reader to [20; 14].”
Convex geodesic bicombings and hyperbolicity (2014, methodology): “(Here we have adopted the notation from [18], whereas [7] uses Kf = {(x, y) ∈ X×X : f(x)+ f(y) = d(x, y)} in place of A(f).”