Universal spaces for \(\mathbb{R}{}\)-trees.

*(English)*Zbl 0787.54036Summary: \(\mathbb{R}\)-trees arise naturally in the study of groups of isometries of hyperbolic space. An \(\mathbb{R}\)-tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. It follows that an \(\mathbb{R}\)-tree is locally arcwise connected, contractible, and one- dimensional. Unique and local arcwise connectivity characterize \(\mathbb{R}\)- trees among metric spaces. A universal \(\mathbb{R}\)-tree would be of interest in attempting to classify the actions of groups of isometries on \(\mathbb{R}\)- trees. It is easy to see that there is no universal \(\mathbb{R}\)-tree. However, we show that there is a universal separable \(\mathbb{R}\)-tree \(T_{\aleph_ 0}\). Moreover, for each cardinal \(\alpha\), \(3 \leq \alpha \leq \aleph_ 0\), there is a space \(T_ \alpha \subset T_{\aleph_ 0}\), universal for separable \(\mathbb{R}\)-trees, whose order of ramification is at most \(\alpha\). We construct a universal smooth dendroid \(D\) such that each separable \(\mathbb{R}\)-tree embeds in \(D\); thus, has a smooth dendroid compactification. For nonseparable \(\mathbb{R}\)-trees, we show that there is an \(\mathbb{R}\)-tree \(X_ \alpha\), such that each \(\mathbb{R}\)-tree of order of ramification at most \(\alpha\) embeds isometrically into \(X_ \alpha\). We also show that each \(\mathbb{R}\)-tree has a compactification into a smooth arboroid (a nonmetric dendroid). We conclude with several examples that show that the characterization of \(\mathbb{R}\)-trees among metric spaces, rather than, say, among first countable spaces, is the best that can be expected.

##### 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) |

54E40 | Special maps on metric spaces |

30F25 | Ideal boundary theory for Riemann surfaces |

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