Universal spaces for R-trees

Academic Article

Abstract

  • 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 Tϗ0. Moreover, for each cardinal α, 3 ≤ α ≤ ϗ0, there is a space Tα ⊂ ϗ0, 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. © 1992 American Mathematical Society.
  • Digital Object Identifier (doi)

    Author List

  • Mayer JC; Nikiel J; Oversteegen LG
  • Start Page

  • 411
  • End Page

  • 432
  • Volume

  • 334
  • Issue

  • 1