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St. Petersburg Mathematical Journal
St. Petersburg Mathematical Journal
ISSN 1547-7371(online) ISSN 1061-0022(print)

 

Hyperbolic dimension of metric spaces


Authors: S. Buyalo and V. Schroeder
Translated by: S. V. Buyalo
Original publication: Algebra i Analiz, tom 19 (2007), nomer 1.
Journal: St. Petersburg Math. J. 19 (2008), 67-76
MSC (2000): Primary 54F45, 53C45
Published electronically: December 12, 2007
MathSciNet review: 2319511
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Abstract: A new quasi-isometry invariant of metric spaces, called the hyperbolic dimension ( $ \operatorname{hypdim}$) is introduced; this is a version of Gromov's asymptotic dimension ( $ \operatorname{asdim}$). The inequality $ \operatorname{hypdim}\le\operatorname{asdim}$ is always fulfilled; however, unlike the asymptotic dimension, $ \operatorname{hypdim}\mathbb{R}^n=0$ for every Euclidean space $ \mathbb{R}^n$ (while $ \operatorname{asdim}\mathbb{R}^n=n$). This invariant possesses the usual properties of dimension such as monotonicity and product theorems. The main result says that the hyperbolic dimension of any Gromov hyperbolic space $ X$ (under mild restrictions) is at least the topological dimension of the boundary at infinity plus 1, $ \operatorname{hypdim} X\ge\dim\partial_{\infty}X+1$. As an application, it is shown that there is no quasi-isometric embedding of the real hyperbolic space $ \operatorname{H}^n$ into the metric product of $ n-1$ metric trees stabilized by any Euclidean factor, $ T_1\times\dots\times T_{n-1}\times\mathbb{R}^m$, $ m\ge 0$.


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Additional Information

S. Buyalo
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: sbuyalo@pdmi.ras.ru

V. Schroeder
Affiliation: Institut für Mathematik, Universität Zürich, Winterthurer Strasse 190, CH-8057, Zürich, Switzerland
Email: vschroed@math.unizh.ch

DOI: http://dx.doi.org/10.1090/S1061-0022-07-00986-7
PII: S 1061-0022(07)00986-7
Keywords: Hyperbolic dimension, Gromov's asymptotic dimension
Received by editor(s): October 10, 2006
Published electronically: December 12, 2007
Additional Notes: The first author was supported by RFBR (grant no. 02-01-00090), by CRDF (grant no. RM1-2381-ST-02), and by SNF (grant no. 20-668 33.01).
The second author was supported by the Swiss National Science Foundation
Dedicated: To Viktor Abramovich Zalgaller
Article copyright: © Copyright 2007 American Mathematical Society