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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
DOI: https://doi.org/10.1090/S0002-9947-1992-1081940-X
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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  • [AB] R. Alperin and H. Bass, Length functions of group actions on $ \Lambda $-trees, Combinatorial Group Theory and Topology (Alta, Utah, 1984), Ann. of Math. Stud., no. 111, Princeton Univ. Press, Princeton, N.J., 1987, pp. 265-378. MR 895622 (89c:20057)
  • [B] H. Bass, $ \Lambda $-trees, Lecture at CBMS Conference, UCLA, Summer 1986.
  • [Be] M. Bestvina, Degenerations of the hyperbolic space, Duke Math. J. 56 (1988), 143-161. MR 932860 (89m:57011)
  • [CE] J. J. Charatonik and C. Eberhart, On smooth dendroids, Fund. Math. 67 (1970), 297-322. MR 0275372 (43:1129)
  • [DJ] K. J. Devlin and H. Johnsbraten, The Souslin problem, Lecture Notes in Math., vol. 405, Springer-Verlag, New York, 1974. MR 0384542 (52:5416)
  • [K] J. L. Kelley, General topology, Springer-Verlag, New York, 1955. MR 0370454 (51:6681)
  • [MO] J. C. Mayer and L. G. Oversteegen, A topological characterization of $ {\mathbf{R}}$-trees, Trans. Amer. Math. Soc. 320 (1991), 395-415. MR 961626 (90k:54031)
  • [vMW] J. van Mill and E. Wattel, Souslin dendrons, Proc. Amer. Math. Soc. 72 (1978), 545-555. MR 509253 (81c:54058)
  • [MoN] L. K. Mohler and J. Nikiel, A universal smooth dendroid answering a question of J. Krasinkiewicz, Houston J. Math. 14 (1988), 535-541. MR 998456 (90h:54037)
  • [Mr] J. W. Morgan, Deformations of algebraic and geometric structures, CBMS Lectures, UCLA, Summer 1986 (preprint).
  • [MrS] J. W. Morgan and P. B. Shalen, Valuations, trees and degenerations of hyperbolic structures. I, Ann. of Math. (2) 120 (1984), 401-476. MR 769158 (86f:57011)
  • [N] J. Nikiel, Topologies on pseudo-trees and applications, Mem. Amer. Math. Soc. No. 416 (1989). MR 988352 (90e:54075)
  • [ST] I. M. Singer and J. A. Thorpe, Lecture notes on elementary topology and geometry, Springer-Verlag, New York, 1967. MR 0213982 (35:4834)
  • [W] R. C. Walker, The Stone-Čech compactification, Springer-Verlag, New York, 1974. MR 0380698 (52:1595)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1992-1081940-X
Keywords: Dendron, dendroid, dendrite, $ {\mathbf{R}}$-tree, universal space, compactification, uniquely arcwise connected, locally arcwise connected
Article copyright: © Copyright 1992 American Mathematical Society

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