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.

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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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2008 09 Feb 2009 14:34 05 Apr 2016 12:57 American Mathematical Society 1061-0022 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. https://doi.org/10.1090/S1061-0022-07-00986-7 http://www.ams.org/mathscinet-getitem?mr=2319511http://arxiv.org/abs/math/0404525v1
Permanent URL: https://doi.org/10.5167/uzh-12698