Universal spaces for $\textbf {R}$-trees

Abstract:

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