Characterization of separable metric 𝑅-trees

Source abstract

An R {\mathbf {R}} -tree ( X , d ) (X,d) is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. R {\mathbf {R}} -trees arise naturally in the study of groups of isometries of hyperbolic space. Two of the authors had previously characterized R {\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 ) (X,r) be a separable metric space. Then the following are equivalent: (1) X X admits an equivalent metric d {\text {d}} such that ( X , d ) (X,d) is an R {\mathbf {R}} -tree . (2) X X is locally arcwise connected and uniquely arcwise connected . The method of proving that (2) implies (1) is to "improve" the metric r 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.