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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Universal spaces for $ {\bf R}$-trees

Authors: John C. Mayer, Jacek Nikiel and Lex G. Oversteegen
Journal: Trans. Amer. Math. Soc. 334 (1992), 411-432
MSC: Primary 54F50; Secondary 30F25, 54E35
MathSciNet review: 1081940
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Abstract: $ {\mathbf{R}}$-trees arise naturally in the study of groups of isometries of hyperbolic space. An $ {\mathbf{R}}$-tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. It follows that an $ {\mathbf{R}}$-tree is locally arcwise connected, contractible, and one-dimensional. Unique and local arcwise connectivity characterize $ {\mathbf{R}}$-trees among metric spaces. A universal $ {\mathbf{R}}$-tree would be of interest in attempting to classify the actions of groups of isometries on $ {\mathbf{R}}$-trees. It is easy to see that there is no universal $ {\mathbf{R}}$-tree. However, we show that there is a universal separable $ {\mathbf{R}}$-tree $ {T_{{\aleph _0}}}$ . Moreover, for each cardinal $ \alpha ,3 \leq \alpha \leq {\aleph _0}$ , there is a space $ {T_\alpha } \subset {T_{{\aleph _0}}}$ , universal for separable $ {\mathbf{R}}$-trees, whose order of ramification is at most $ \alpha $ . We construct a universal smooth dendroid $ D$ such that each separable $ {\mathbf{R}}$-tree embeds in $ D$ ; thus, has a smooth dendroid compactification. For nonseparable $ {\mathbf{R}}$-trees, we show that there is an $ {\mathbf{R}}$-tree $ {X_\alpha }$ , such that each $ {\mathbf{R}}$-tree of order of ramification at most $ \alpha $ embeds isometrically into $ {X_\alpha }$ . We also show that each $ {\mathbf{R}}$-tree has a compactification into a smooth arboroid (a nonmetric dendroid). We conclude with several examples that show that the characterization of $ {\mathbf{R}}$-trees among metric spaces, rather than, say, among first countable spaces, is the best that can be expected.

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Keywords: Dendron, dendroid, dendrite, $ {\mathbf{R}}$-tree, universal space, compactification, uniquely arcwise connected, locally arcwise connected
Article copyright: © Copyright 1992 American Mathematical Society