Universal Spaces for R-trees

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 7n0 . Moreover, for each cardinal a, 3 < a < H0 , there is a space Ta C 7n0 , universal for separable R-trees, whose order of ramification is at most a . 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 Xa , such that each R-tree of order of ramification at most a embeds isometrically into Xa . 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.