Hyperbolic dimension of metric spaces
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S. Buyalo and V. Schroeder
Translated by: S. V. Buyalo - St. Petersburg Math. J. 19 (2008), 67-76
- DOI: https://doi.org/10.1090/S1061-0022-07-00986-7
- Published electronically: December 12, 2007
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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, $T_1\times \dots \times T_{n-1}\times \mathbb {R}^m$, $m\ge 0$.References
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Bibliographic 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
- MR Author ID: 157030
- Email: vschroed@math.unizh.ch
- 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 - © Copyright 2007 American Mathematical Society
- Journal: St. Petersburg Math. J. 19 (2008), 67-76
- MSC (2000): Primary 54F45, 53C45
- DOI: https://doi.org/10.1090/S1061-0022-07-00986-7
- MathSciNet review: 2319511
Dedicated: To Viktor Abramovich Zalgaller