A topological characterization of $\textbf {R}$-trees

Abstract:

$\mathbf {R}$-trees arise naturally in the study of groups of isometries of hyperbolic space. An $\mathbf {R}$-tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals $\mathbf {R}$. Actions on $\mathbf {R}$-trees can be viewed as ideal points in the compactification of groups of isometries. As such they have applications to the study of hyperbolic manifolds. Our concern in this paper, however, is with the topological characterization of $\mathbf {R}$-trees. Our main theorem is the following: Let $(X,p)$ be a metric space. Then $X$ is uniquely arcwise connected and locally arcwise connected if, and only if, $X$ admits a compatible metric $d$ such that $(X,d)$ is an $\mathbf {R}$-tree. Essentially, we show how to put a convex metric on a uniquely arcwise connected, locally arcwise connected, metrizable space.