Characterization of separable metric -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 -tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. -trees arise naturally in the study of groups of isometries of hyperbolic space. Two of the authors had previously characterized -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 be a separable metric space. Then the following are equivalent:
(1) admits an equivalent metric such that is an -tree.
(2) is locally arcwise connected and uniquely arcwise connected. The method of proving that (2) implies (1) is to "improve" the metric 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: -tree, convex metric, uniquely arcwise connected, locally arcwise connected
Article copyright: © Copyright 1992 American Mathematical Society