CHARACTERIZATION OF SEPARABLE METRIC R-TREES

An R-tree (X, d) is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. 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 such that (X, d) is an 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.