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Universal spaces for $$\mathbb{R}{}$$-trees. (English) Zbl 0787.54036
Summary: $$\mathbb{R}$$-trees arise naturally in the study of groups of isometries of hyperbolic space. An $$\mathbb{R}$$-tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. It follows that an $$\mathbb{R}$$-tree is locally arcwise connected, contractible, and one- dimensional. Unique and local arcwise connectivity characterize $$\mathbb{R}$$- trees among metric spaces. A universal $$\mathbb{R}$$-tree would be of interest in attempting to classify the actions of groups of isometries on $$\mathbb{R}$$- trees. It is easy to see that there is no universal $$\mathbb{R}$$-tree. However, we show that there is a universal separable $$\mathbb{R}$$-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 $$\mathbb{R}$$-trees, whose order of ramification is at most $$\alpha$$. We construct a universal smooth dendroid $$D$$ such that each separable $$\mathbb{R}$$-tree embeds in $$D$$; thus, has a smooth dendroid compactification. For nonseparable $$\mathbb{R}$$-trees, we show that there is an $$\mathbb{R}$$-tree $$X_ \alpha$$, such that each $$\mathbb{R}$$-tree of order of ramification at most $$\alpha$$ embeds isometrically into $$X_ \alpha$$. We also show that each $$\mathbb{R}$$-tree has a compactification into a smooth arboroid (a nonmetric dendroid). We conclude with several examples that show that the characterization of $$\mathbb{R}$$-trees among metric spaces, rather than, say, among first countable spaces, is the best that can be expected.

##### MSC:
 54F65 Topological characterizations of particular spaces 54F50 Topological spaces of dimension $$\leq 1$$; curves, dendrites 54E35 Metric spaces, metrizability 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 54D05 Connected and locally connected spaces (general aspects) 54E40 Special maps on metric spaces 30F25 Ideal boundary theory for Riemann surfaces 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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