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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Asymptotic dimension of a hyperbolic space and capacity dimension of its boundary at infinity

Author: S. Buyalo
Translated by: the author
Original publication: Algebra i Analiz, tom 17 (2005), nomer 2.
Journal: St. Petersburg Math. J. 17 (2006), 267-283
MSC (2000): Primary 53B99
Published electronically: February 10, 2006
MathSciNet review: 2159584
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Abstract | References | Similar Articles | Additional Information

Abstract: A quasisymmetry invariant of a metric space $ Z$ (called the capacity dimension, $ \operatorname{cdim} Z$) is introduced. The main result says that the asymptotic dimension of a visual Gromov hyperbolic space $ X$ is at most the capacity dimension of its boundary at infinity plus 1, $ \operatorname{asdim} X \le \operatorname{cdim} \partial_\infty X+1$.

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

  • [As] Patrice Assouad, Sur la distance de Nagata, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 1, 31–34 (French, with English summary). MR 651069
  • [BD] G. Bell and A. Dranishnikov, On asymptotic dimension of groups acting on trees, arXiv:math.GR/0111087.
  • [BoS] M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), no. 2, 266–306. MR 1771428,
  • [BS] S. Buyalo and V. Schroeder, Hyperbolic dimension of metric spaces, arXiv:math.GT/0404525.
  • [Gr] M. Gromov, Asymptotic invariants of infinite groups, Geometric group theory, Vol. 2 (Sussex, 1991) London Math. Soc. Lecture Note Ser., vol. 182, Cambridge Univ. Press, Cambridge, 1993, pp. 1–295. MR 1253544
  • [He] Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917
  • [LS] U. Lang and T. Schlichenmaier, Nagata dimension, quasisymmetric embeddings and Lipschitz extensions, arXiv:math.MG/0410048.
  • [Sp] Edwin H. Spanier, Algebraic topology, Springer-Verlag, New York-Berlin, 1981. Corrected reprint. MR 666554

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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

Keywords: Visual Gromov hyperbolic space
Received by editor(s): November 1, 2004
Published electronically: February 10, 2006
Additional Notes: The author was supported by RFBR (grant no. 02-01-00090).
Article copyright: © Copyright 2006 American Mathematical Society