Hyperbolizing hyperspaces
type: Item
instance of: scholarly article
title: Hyperbolizing hyperspaces
zbMATH Open document ID: 1220.30075
author: Zair Ibragimov
published in: Michigan Mathematical Journal
publication date: 2011-05-23T00:00:00Z
review text: The aim of this paper is to establish connections between a metric space \(X\) and the large-scale geometry (in the sense of Gromov) of the hyperspace \(\mathcal H(X)\) of its non-degenerate closed bounded subsets, and to study the mappings on \(X\) in terms of the induced mappings on \(\mathcal H(X)\). The metric space \(X\) can be identified with the boundary of \(\mathcal H(X)\) when the latter is equipped with the Hausdorff metric, but stronger relationships between \(X\) and \(\mathcal H(X)\) are obtained when the hyperspace \(\mathcal H(X)\) is hyperbolized and the space \(X\) is identified with its boundary at infinity: a priori weak conditions on \(\mathcal H(X)\) are strengthened at the boundary at infinity. The basic tool for studying such relationships is Gromov's theory of negatively curved spaces. Let us consider Euclidean space \(\mathbb R^n\). The one-point compactification \(\overline{\mathbb R^n}\) is equipped with the chordal metric, which is Möbius equivalent to the Euclidean metric when restricted to \({\mathbb R^n}\). The space \(\overline{\mathbb R^n}\) can be identified with the ideal boundary of the hyperbolic space \(\mathbb H^{n+1}\) and, as such, it inherits a family of visual metrics from \(\mathbb H^{n+1}\). In fact, the chordal metric is one such visual metric. The Möbius transformations of \(\overline{\mathbb R^n}\) can be extended to isometries of \(\mathbb H^{n+1}\) by the Poincaré extension. The quasisymmetric maps of \(\mathbb R^n\) can be extended to bi-Lipschitz maps of \(\mathbb H^{n+1}\) by the Tukia-Väisälä extension operator and to quasi-isometries of \(\mathbb H^{n+1}\) by the Tukia extension operator. In this paper, the author shows that connections analogous to those just described between \(\mathbb R^n\) and \(\mathbb H^{n+1}\) can be established between a more general metric space and its hyperspace. More precisely, the author replaces the space \(\mathbb R^n\) with an arbitrary complete perfect metric space \(X\). The o…
Mathematics Subject Classification ID: 30L99
Mathematics Subject Classification ID: 30F45
Mathematics Subject Classification ID: 32Q40
Mathematics Subject Classification ID: 32Q45
Mathematics Subject Classification ID: 54B20
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 5899032
zbMATH Keywords: Gromov hyperbolic spaces
zbMATH Keywords: chordal metric
zbMATH Keywords: quasi-isometric embeddings of hypersurfaces
zbMATH Keywords: hyperbolizing hyperspaces
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1301586312
OpenAlex ID: W4247479642
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DOI: 10.1307/MMJ/1301586312
description: scientific article; zbMATH DE number 5899032