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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

A topological characterization of $ {\bf R}$-trees


Authors: John C. Mayer and Lex G. Oversteegen
Journal: Trans. Amer. Math. Soc. 320 (1990), 395-415
MSC: Primary 54D05; Secondary 54E35, 54F50, 54H99
MathSciNet review: 961626
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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 $ \mathbf{R}$. Actions on $ \mathbf{R}$-trees can be viewed as ideal points in the compactification of groups of isometries. As such they have applications to the study of hyperbolic manifolds. Our concern in this paper, however, is with the topological characterization of $ \mathbf{R}$-trees. Our main theorem is the following: Let $ (X,p)$ be a metric space. Then $ X$ is uniquely arcwise connected and locally arcwise connected if, and only if, $ X$ admits a compatible metric $ d$ such that $ (X,d)$ is an $ \mathbf{R}$-tree. Essentially, we show how to put a convex metric on a uniquely arcwise connected, locally arcwise connected, metrizable space.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1990-0961626-8
PII: S 0002-9947(1990)0961626-8
Keywords: Tree, $ \mathbf{R}$-tree, dendroid, smooth dendroid, locally connected, arcwise connected, locally arcwise connected, uniquely arcwise connected, metrization, equivalent metric, compatible metric, convex metric, radially convex metric, $ \mathbf{R}$-tree metric
Article copyright: © Copyright 1990 American Mathematical Society