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St. Petersburg Mathematical Journal

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



Dimensions of products of hyperbolic spaces

Author: N. Lebedeva
Translated by: the author
Original publication: Algebra i Analiz, tom 19 (2007), nomer 1.
Journal: St. Petersburg Math. J. 19 (2008), 107-124
MSC (2000): Primary 54F45
Published electronically: December 17, 2007
MathSciNet review: 2319513
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Abstract | References | Similar Articles | Additional Information

Abstract: Estimates on asymptotic dimension are given for products of general hyperbolic spaces, with applications to hyperbolic groups. Examples are presented where strict inequality occurs in the product theorem for the asymptotic dimension in the class of hyperbolic groups and in the product theorem for the hyperbolic dimension. It is proved that $ \mathbb{R}$ is dimensionally full for the asymptotic dimension in the class of hyperbolic groups.

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  • [BD] G. Bell and A. Dranishnikov, On asymptotic dimension of groups, Algebr. Geom. Topol. 1 (2001), 57-71. MR 1808331 (2001m:20062)
  • [BoS] M. Bonk and O. Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000), no. 2, 266-306. MR 1771428 (2001g:53077)
  • [Bu1] S. V. Buyalo, Asymptotic dimension of a hyperbolic space and capacity dimension of its boundary at infinity, Algebra i Analiz 17 (2005), no. 2, 70-95; English transl., St. Petersburg Math. J. 17 (2006), no. 2, 267-283. MR 2159584 (2006d:31009)
  • [Bu2] -, Capacity dimension and embedding of hyperbolic spaces into a product of trees, Algebra i Analiz 17 (2005), no. 4, 42-58; English transl., St. Petersburg Math. J. 17 (2006), no. 4, 581-591. MR 2173936 (2006e:31008)
  • [BS1] S. Buyalo and V. Schroeder, Embedding of hyperbolic spaces in the product of trees, Geom. Dedicata 113 (2005), 75-93. MR 2171299 (2006f:53055)
  • [BS2] -, Hyperbolic dimension of metric spaces, arXive:math. GT/0404525; Algebra i Analiz 19 (2007), no. 1, 93-108; English transl. in St. Petersburg Math. J. 19 (2008), no. 1. MR 2319511
  • [BS3] -, Elements of asymptotic geometry (in preparation).
  • [BL] S. Buyalo and N. Lebedeva, Dimensions of locally and asymptotically self-similar spaces, arXiv:math.GT/0509433; Algebra i Analiz 19 (2007), no. 1, 60-92; English transl. in St. Petersburg Math. J. 19 (2008), no. 1. MR 2319510
  • [Dr1] A. Dranishnikov, On the virtual cohomological dimensions of Coxeter groups, Proc. Amer. Math. Soc. 125 (1997), 1885-1891. MR 1422863 (98d:55001)
  • [Dr2] -, Boundaries of Coxeter groups and simplicial complexes with given links, J. Pure Appl. Algebra 137 (1999), 139-151. MR 1684267(2000d:20069)
  • [Dr3] -, Open problems in asymptotic dimension theory, Preprint, 2006.
  • [Dr4] -, Cohomological approach to asymptotic dimension, arXive:math.MG/0608215.
  • [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)

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Additional Information

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

Keywords: Asymptotic dimension, hyperbolic groups, linearly controlled dimension, quasi-isometry invariants.
Received by editor(s): June 19, 2007
Published electronically: December 17, 2007
Additional Notes: Supported by RFBR (grant no. 05-01-00939)
Dedicated: To dear Viktor Abramovich Zalgaller on the occasion of his 85th birthday
Article copyright: © Copyright 2007 American Mathematical Society

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