Characterization of separable metric $\textbf {R}$-trees
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- by J. C. Mayer, L. K. Mohler, L. G. Oversteegen and E. D. Tymchatyn PDF
- Proc. Amer. Math. Soc. 115 (1992), 257-264 Request permission
Abstract:
An ${\mathbf {R}}$-tree $(X,d)$ is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. ${\mathbf {R}}$-trees arise naturally in the study of groups of isometries of hyperbolic space. Two of the authors had previously characterized ${\mathbf {R}}$-trees topologically among metric spaces. The purpose of this paper is to provide a simpler proof of this characterization for separable metric spaces. The main theorem is the following: Let $(X,r)$ be a separable metric space. Then the following are equivalent: (1) $X$ admits an equivalent metric ${\text {d}}$ such that $(X,d)$ is an ${\mathbf {R}}$-tree. (2) $X$ is locally arcwise connected and uniquely arcwise connected. The method of proving that (2) implies (1) is to "improve" the metric $r$ through a sequence of equivalent metrics of which the first is monotone on arcs, the second is strictly monotone on arcs, and the last is convex, as desired.References
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Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 115 (1992), 257-264
- MSC: Primary 54F50; Secondary 54E35, 54F65
- DOI: https://doi.org/10.1090/S0002-9939-1992-1124147-5
- MathSciNet review: 1124147