Trees and ultrametric spaces: A categorical equivalence
type: Item
instance of: scholarly article
title: Trees and ultrametric spaces: A categorical equivalence
zbMATH Open document ID: 1061.57021
author: Bruce Hughes
published in: Advances in Mathematics
publication date: 2004-11-26T00:00:00Z
review text: 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.
reviewed by: Maria Rita Casali
Mathematics Subject Classification ID: 57M99
Mathematics Subject Classification ID: 05C05
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 2119096
zbMATH Keywords: tree
zbMATH Keywords: ultrametric space
zbMATH Keywords: end space
zbMATH Keywords: isometry at infinity
zbMATH Keywords: local similarity
zbMATH Keywords: local isometry
MaRDI profile type: Publication
full work available at URL: j.aim.2003.11.008
OpenAlex ID: W2056591797
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DOI: 10.1016/J.AIM.2003.11.008
description: scientific article; zbMATH DE number 2119096