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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Characterization of separable metric $ {\bf R}$-trees

Authors: J. C. Mayer, L. K. Mohler, L. G. Oversteegen and E. D. Tymchatyn
Journal: Proc. Amer. Math. Soc. 115 (1992), 257-264
MSC: Primary 54F50; Secondary 54E35, 54F65
MathSciNet review: 1124147
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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.

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Keywords: $ {\text{R}}$-tree, convex metric, uniquely arcwise connected, locally arcwise connected
Article copyright: © Copyright 1992 American Mathematical Society

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