Characterization of separable metric R-trees

Academic Article

Abstract

  • An R-tree (X, d) is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of thereals. R-trees arise naturally in the study of groups of isometries of hyperbolic space. Two of the authors had previously characterized 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 d suchthat (X, d) is an R-tree. (2)X is locallyarcwise 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. © 1992 American Mathematical Society.
  • Digital Object Identifier (doi)

    Author List

  • Mayer JC; Mohler LK; Oversteegen LG; Tymchatyn ED
  • Start Page

  • 257
  • End Page

  • 264
  • Volume

  • 115
  • Issue

  • 1