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

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.