Trees and ultrametric spaces: A categorical equivalence
A well-known correspondence between infinite trees and ultrametric spaces exists, which takes into account the end space of the tree. The present paper deeply analyzes this correspondence in terms of categories, by introducing suitable categories that capture the geometry of trees at infinity and the micro-geometry of ultrametric spaces. The main theorem (resp. its most important corollary) states that {there is an equivalence from the category of geodesically complete, rooted \(\mathbb R\)-trees and equivalence classes of isometries (resp. uniform isometries) at infinity, to the category of complete ultrametric spaces of finite diameter and locally similarity (resp. locally uniform similarity) equivalences.} Note that many related results already exist in literature: see, for example [\textit{H. J. Baues} and \textit{A. Quintero}, Infinite homotopy theory. K-Monographs in Mathematics 6 (2001; Zbl 0983.55001); \textit{F. Choucroun}, Geom. Dedicata 53, 69--74 (1994; Zbl 0867.54032); Progress in Mathematics 83, 103--116 (1990; Zbl 0731.20025); \textit{R. I. Grigorchuk, V. V. Nekrashevich} and \textit{V.I. Sushchanskii}, Proc. Steklov Inst. Math. 231, 128--203 (2000; Zbl 1155.37311)]. However, the approach performed in the present paper appears to be really new and interesting, because of the combination of the following three elements: not only are the objects of the two categories in one-to-one correspondence, but the categories themselves are shown to be equivalent; the geometry of the tree, rather than its topology, is studied (in fact, the natural metric on the end space is emphasized when passing from trees to an ideal space at infinity using end theory); the results are set in \(\mathbb R\)-trees rather than more classical type of trees.
scientific article; zbMATH DE number 2119096