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The hyperbolic dimension of metric spaces


Buyalo, S; Schroeder, V (2008). The hyperbolic dimension of metric spaces. St. Petersburg Mathematical Journal, 19(1):67-76.

Abstract

We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, hypdim, which is a version of the Gromov's asymptotic dimension, asdim. The hyperbolic dimension is at most the asymptotic dimension, however, unlike the asymptotic dimension, the hyperbolic dimension of any Euclidean space R^n is zero (while asdim R^n=n.) This invariant possesses usual properties of dimension like monotonicity and product theorems. Our main result says that the hyperbolic dimension of any Gromov hyperbolic space X (with mild restrictions) is at least the topological dimension of the boundary at infinity plus 1. As an application we obtain that there is no quasi-isometric embedding of the real hyperbolic space H^n into the (n-1)-fold metric product of metric trees stabilized by any Euclidean factor.

Abstract

We introduce a new quasi-isometry invariant of metric spaces called the hyperbolic dimension, hypdim, which is a version of the Gromov's asymptotic dimension, asdim. The hyperbolic dimension is at most the asymptotic dimension, however, unlike the asymptotic dimension, the hyperbolic dimension of any Euclidean space R^n is zero (while asdim R^n=n.) This invariant possesses usual properties of dimension like monotonicity and product theorems. Our main result says that the hyperbolic dimension of any Gromov hyperbolic space X (with mild restrictions) is at least the topological dimension of the boundary at infinity plus 1. As an application we obtain that there is no quasi-isometric embedding of the real hyperbolic space H^n into the (n-1)-fold metric product of metric trees stabilized by any Euclidean factor.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:510 Mathematics
Language:English
Date:2008
Deposited On:09 Feb 2009 14:34
Last Modified:05 Apr 2016 12:57
Publisher:American Mathematical Society
ISSN:1061-0022
Additional Information:First published in Buyalo, S; Schroeder, V (2008). The hyperbolic dimension of metric spaces. St.Petersburg Mathematical Journal, 19(1):67-76, published by the American Mathematical Society.
Publisher DOI:https://doi.org/10.1090/S1061-0022-07-00986-7
Related URLs:http://www.ams.org/mathscinet-getitem?mr=2319511
http://arxiv.org/abs/math/0404525v1

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