Universal spaces for trees
Authors:
John C. Mayer, Jacek Nikiel and Lex G. Oversteegen
Journal:
Trans. Amer. Math. Soc. 334 (1992), 411432
MSC:
Primary 54F50; Secondary 30F25, 54E35
MathSciNet review:
1081940
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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 onedimensional. 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:
http://dx.doi.org/10.1090/S0002994719921081940X
PII:
S 00029947(1992)1081940X
Keywords:
Dendron,
dendroid,
dendrite,
tree,
universal space,
compactification,
uniquely arcwise connected,
locally arcwise connected
Article copyright:
© Copyright 1992 American Mathematical Society
