Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

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

 
 

 

Capacity dimension and embedding of hyperbolic spaces into a product of trees


Author: S. Buyalo
Translated by: the author
Original publication: Algebra i Analiz, tom 17 (2005), nomer 4.
Journal: St. Petersburg Math. J. 17 (2006), 581-591
MSC (2000): Primary 51M10
DOI: https://doi.org/10.1090/S1061-0022-06-00921-6
Published electronically: May 3, 2006
MathSciNet review: 2173936
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is proved that every visual Gromov hyperbolic space $ X$ whose boundary at infinity has finite capacity dimension, $ \operatorname{cdim}(\partial_{\infty} X)<\infty$, admits a quasiisometric embedding into an $ n$-fold product of metric trees with $ n=\operatorname{cdim}(\partial_{\infty} X)+1$.


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

  • [As] P. Assouad, Plongements lipschitziens dans $ R^n$, Bull. Soc. Math. France 111 (1983), 429-448. MR 0763553 (86f:54050)
  • [BoS] M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), no. 2, 266-306. MR 1771428 (2001g:53077)
  • [Bu] S. V. Buyalo, Asymptotic dimension of the hyperbolic space and the capacity dimension of its boundary at infinity, Algebra i Analiz 17 (2005), no. 2, 70-95; English transl. in St. Petersburg Math. J. 17 (2006), no. 2. MR 2159584 (2006d:31009)
  • [BS1] S. Buyalo and V. Schroeder, Embedding of hyperbolic spaces in the product of trees, arXive:math. GT/0311524 (2003).
  • [BS2] -, Hyperbolic dimension of metric spaces, arXive:math. GT/0404525 (2004).
  • [Dr] A. Dranishnikov, On hypersphericity of manifolds with finite asymptotic dimension, Trans. Amer. Math. Soc. 355 (2003), 155-167. MR 1928082 (2003g:53055)
  • [DZ] A. Dranishnikov and M. Zarichnyi, Universal spaces for asymptotic dimension, Topology Appl. 140 (2004), 203-225; arXive:math. GT/0211069 (2002). MR 2074917 (2005e:54032)
  • [He] J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917 (2002c:30028)
  • [LS] U. Lang and T. Schlichenmaier, Nagata dimension, quasisymmetric embeddings and Lipschitz extensions, arXive:math. MG/0410048 (2004).
  • [Ro] J. Roe, Lectures on coarse geometry, Univ. Lecture Ser., vol. 31, Amer. Math. Soc., Providence, RI, 2003. MR 2007488 (2004g:53050)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 51M10

Retrieve articles in all journals with MSC (2000): 51M10


Additional Information

S. Buyalo
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

DOI: https://doi.org/10.1090/S1061-0022-06-00921-6
Keywords: Visual Gromov hyperbolic space, asymptotic dimension, capacity dimension
Received by editor(s): March 9, 2005
Published electronically: May 3, 2006
Additional Notes: Partially supported by RFBR (grant no. 05-01-00939) and by NSH (grant no. 1914.2003.1)
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society