Remote Access 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
DOI: https://doi.org/10.1090/S1061-0022-07-00986-7
Published electronically: December 12, 2007
MathSciNet review: 2319511
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [Al] P. Alexandroff, Dimensionstheorie. Ein Beitrag zur Geometrie der abgeschlossenen Mengen, Math. Ann. 106 (1932), 161-238.
  • [BD] G. Bell and A. Dranishnikov, On asymptotic dimension of groups acting on trees, Algebr. Geom. Topol. 1 (2001), 57-71 (electronic). MR 1808331 (2001m:20062)
  • [BM] M. Bestvina and G. Mess, The boundary of negatively curved groups, J. Amer. Math. Soc. 4 (1991), no. 3, 469-481. MR 1096169 (93j:20076)
  • [BoS] M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), no. 2, 266-306. MR 1771428 (2001g:53077)
  • [BDS] S. Buyalo, A. Dranishnikov, and V. Schroeder, Embedding of hyperbolic groups into products of binary trees, Invent. Math. 169 (2007), no. 1, 153-192. MR 2308852
  • [BS1] S. Buyalo and V. Schroeder, Hyperbolic rank and subexponential corank of metric spaces, Geom. Funct. Anal. 12 (2002), 293-306. MR 1911661 (2003e:53045)
  • [BS2] -, Embedding of hyperbolic spaces in the product of trees, Geom. Dedicata 113 (2005), 75-93. MR 2171299 (2006f:53055)
  • [BS3] -, Elements of asymptotic geometry, EMS Monographs in Mathematics, 2007. European Mathematical Society (EMS), Zürich, 2007. MR 2327160
  • [DJ] A. Dranishnikov and T. Januszkiewicz, Every Coxeter group acts amenably on a compact space, Proceedings of the 1999 Topology and Dynamics Conference (Salt Lake City, UT), Topology Proc. 24 (1999), Spring, 135-141. MR 1802681 (2001k:20082)
  • [Gr] M. Gromov, Asymptotic invariants of infinite groups, Geometric Group Theory, Vol. 2 (Sussex, 1991) (G. A. Niblo, M. A. Roller, eds.), London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1-295. MR 1253544 (95m:20041)
  • [HW] W. Hurewicz and H. Wallman, Dimension theory, Princeton Math. Ser., vol. 4, Princeton Univ. Press, Princeton, NJ, 1941. MR 0006493 (3:312b)
  • [JS] T. Januszkiewicz and J. Swiatkowski, Hyperbolic Coxeter groups of large dimension, Comment. Math. Helv. 78 (2003), 555-583. MR 1998394 (2004h:20058)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 54F45, 53C45

Retrieve articles in all journals with MSC (2000): 54F45, 53C45


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: https://doi.org/10.1090/S1061-0022-07-00986-7
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

American Mathematical Society