Universal Spaces for R-Trees

• Published in: Transactions of the American Mathematical Society 1992-11, Vol.334 (1), p.411-432, Article 411
• Main Authors: Mayer, John C., Nikiel, Jacek, Oversteegen, Lex G.
• Format: Article
• Language: English
• Subjects: Compactification; Dendrimers; Dendrites; Exact sciences and technology; First countable; General topology; Mathematical theorems; Mathematics; Numbers; Sciences and techniques of general use; Topological theorems; Topology; Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds; Vertices
• ISSN: 0002-9947; 1088-6850
• DOI: 10.2307/2153989

R-trees arise naturally in the study of groups of isometries of hyperbolic space. An 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 R-tree is locally arcwise connected, contractible, and one-dimensional. Unique and local arcwise connectivity characterize R-trees among metric spaces. A universal R-tree would be of interest in attempting to classify the actions of groups of isometries on R-trees. It is easy to see that there is no universal R-tree. However, we show that there is a universal separable R-tree Tℵ0 . Moreover, for each cardinal α, 3 ≤ α ≤ ℵ0, there is a space$T_\alpha \subset T_{\aleph_0}$, universal for separable R-trees, whose order of ramification is at most α. We construct a universal smooth dendroid D such that each separable R-tree embeds in D; thus, has a smooth dendroid compactification. For nonseparable R-trees, we show that there is an R-tree Xα, such that each R-tree of order of ramification at most α embeds isometrically into Xα. We also show that each R-tree has a compactification into a smooth arboroid (a nonmetric dendroid). We conclude with several examples that show that the characterization of R-trees among metric spaces, rather than, say, among first countable spaces, is the best that can be expected.