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Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces. (English) Zbl 0562.54041
A metric space Y is injective if every mapping which increases no distance from a subspace of any metric space X to Y can be extended, increasing no distance, over X. In 1964, J. R. Isbell constructed an injective envelope \(T_ X\) of a metric space X [see Comment. Math. Helv. 39, 65-76 (1964; Zbl 0151.302)]. The present paper brings a thorough analysis of this construction and applications of it to (1) the existence of embeddings of metric spaces into trees, (2) optimal graphs realizing a metric space, and (3) the cohomological dimensions of groups with specific length functions. In more detail, a metric space X is a tree if for any two elements x,y\(\in X\) there is - up to a parametrization - only one injective continuous map \(h: [0,1]\to X\) such that \(h(0)=x\) and \(h(1)=y\). Now, \(T_ X\) tests the embeddability of X into a tree, i.e. X is a subspace of a tree if and only if \(T_ X\) is a tree. The relation of \(T_ X\) to optimal realizations of X by networks is a little weaker: any optimal realization of X is contained in \(T_ X\).
Reviewer: J.Rosicky

54E35 Metric spaces, metrizability
54C20 Extension of maps
05C99 Graph theory
20J99 Connections of group theory with homological algebra and category theory
Full Text: DOI
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