Universal spaces for -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

DOI:
https://doi.org/10.1090/S0002-9947-1992-1081940-X

MathSciNet review:
1081940

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Abstract | References | Similar Articles | Additional Information

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

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1992-1081940-X

Keywords:
Dendron,
dendroid,
dendrite,
-tree,
universal space,
compactification,
uniquely arcwise connected,
locally arcwise connected

Article copyright:
© Copyright 1992
American Mathematical Society