Universal spaces for 𝑅-trees

Abstract

R {\mathbf {R}} -trees arise naturally in the study of groups of isometries of hyperbolic space. An R {\mathbf {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 {\mathbf {R}} -tree is locally arcwise connected, contractible, and one-dimensional. Unique and local arcwise connectivity characterize R {\mathbf {R}} -trees among metric spaces. A universal R {\mathbf {R}} -tree would be of interest in attempting to classify the actions of groups of isometries on R {\mathbf {R}} -trees. It is easy to see that there is no universal R {\mathbf {R}} -tree. However, we show that there is a universal separable R {\mathbf {R}} -tree T ℵ 0 {T_{{\aleph _0}}} . Moreover, for each cardinal α , 3 ≤ α ≤ ℵ 0 \alpha ,3 \leq \alpha \leq {\aleph _0} , there is a space T α ⊂ T ℵ 0 {T_\alpha } \subset {T_{{\aleph _0}}} , universal for separable R {\mathbf {R}} -trees, whose order of ramification is at most α \alpha . We construct a universal smooth dendroid D D such that each separable R {\mathbf {R}} -tree embeds in D D ; thus, has a smooth dendroid compactification. For nonseparable R {\mathbf {R}} -trees, we show that there is an R {\mathbf {R}} -tree X α {X_\alpha } , such that each R {\mathbf {R}} -tree of order of ramification at most α \alpha embeds isometrically into X α {X_\alpha } . We also show that each R {\mathbf {R}} -tree has a compactification into a smooth arboroid (a nonmetric dendroid). We conclude with several examples that show that the characterization of R {\mathbf {R}} -trees among metric spaces, rather than, say, among first countable spaces, is the best that can be expected.