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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 $h:\left[0,1\right]\to X$ such that $h\left(0\right)=x$ and $h\left(1\right)=y$. 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

##### MSC:
 54E35 Metric spaces, metrizability 54C20 Extension of maps on topological spaces 05C99 Graph theory 20J99 Connections of group theory with homological algebra