Characterization of separable metric R-trees.

*(English)*Zbl 0754.54026Summary: An \(\mathbb{R}\)-tree \((X,d)\) is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. \(\mathbb{R}\)-trees arise naturally in the study of groups of isometries of hyperbolic space. The first and the third of the authors [in Trans. Am. Math. Soc. 320, 395-415 (1990; Zbl 0729.54008)] had previously characterized \(\mathbb{R}\)-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:
\[
X \text{ admits an equivalent metric } d \text{ such that }(X,d)\text{ is an }\mathbb{R}\text{- tree.}\tag{1}
\]
\[
X \text{ is locally arcwise connected and uniquely arcwise connected.}\tag{2}
\]
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.

##### MSC:

54F65 | Topological characterizations of particular spaces |

54F50 | Topological spaces of dimension \(\leq 1\); curves, dendrites |

54E35 | Metric spaces, metrizability |

54F05 | Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces |

54D05 | Connected and locally connected spaces (general aspects) |

30F25 | Ideal boundary theory for Riemann surfaces |

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |