Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

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

 
 

 

Linearly controlled asymptotic dimension of the fundamental group of a graph-manifold


Author: A. Smirnov
Translated by: the author
Original publication: Algebra i Analiz, tom 22 (2010), nomer 2.
Journal: St. Petersburg Math. J. 22 (2011), 307-319
MSC (2010): Primary 57M50, 55M10; Secondary 05C05, 20F69
DOI: https://doi.org/10.1090/S1061-0022-2011-01142-2
Published electronically: February 8, 2011
MathSciNet review: 2668128
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the estimate $ \ell$-$ \operatorname{asdim} \pi_1(M)\leq 7$ for the linearly controlled asymptotic dimension of the fundamental group of any 3-dimensional graph-manifold $ M$. As applications, we show that the universal cover $ \widetilde{M}$ of $ M$ is an absolute Lipschitz retract and admits a quasisymmetric embedding into the product of 8 metric trees.


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

  • 1. P. Assouad, Sur la distance de Nagata, C. R. Acad. Sci. Paris Sér. I Math. 294 (1982), no. 1, 31-34. MR 0651069 (83b:54034)
  • 2. J. A. Behrstock and W. D. Neumann, Quasi-isometric classification of graph manifold groups, Duke Math. J. 141 (2008), no. 2, 217-240. MR 2376814 (2009c:20070)
  • 3. G. Bell and A. Dranishnikov, On asymptotic dimension of groups acting on trees, Geom. Dedicata 103 (2004), no. 1, 89-101. MR 2034954 (2005b:20078)
  • 4. M. R. Bridson and A. Haefliger, Metric spaces of nonpositive curvature, Grundlehren Math. Wiss., Bd. 319, Springer-Verlag, Berlin, 1999. MR 1744486 (2000k:53038)
  • 5. D. Yu. Burago, Yu. D. Burago, and S. V. Ivanov, A course of metric geometry, Inst. Kompyuter. Issled., Moscow-Izhevsk, 2005. (Russian)
  • 6. S. V. Buyalo, Asymptotic dimension of a hyperbolic space and the 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)
  • 7. A. Dranishnikov and M. Zarichnyi, Universal spaces for asymptotic dimension, Topology Appl. 140 (2004), no. 2-3, 203-225. MR 2074917 (2005e:54032)
  • 8. 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 (95m:20041)
  • 9. U. Lang and T. Schlichenmaier, Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions, Int. Math. Res. Not. 2005, no. 58, 3625-3655. MR 2200122 (2006m:53061)
  • 10. J. Roe, Lectures on coarse geometry, Univ. Lecture Ser., vol. 31, Amer. Math. Soc., Providence, RI, 2003. MR 2007488 (2004g:53050)
  • 11. G. Yu, The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. (2) 147 (1998), no. 2, 325-355. MR 1626745 (99k:57072)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 57M50, 55M10, 05C05, 20F69

Retrieve articles in all journals with MSC (2010): 57M50, 55M10, 05C05, 20F69


Additional Information

A. Smirnov
Affiliation: Mathematics and Mechanics Department, St. Petersburg State University, 28 Universitetskii Prospekt, Peterhoff, St. Petersburg 198504, Russia
Email: alvismi@gmail.com

DOI: https://doi.org/10.1090/S1061-0022-2011-01142-2
Keywords: Graph-manifold, asymptotic dimension
Received by editor(s): April 23, 2009
Published electronically: February 8, 2011
Article copyright: © Copyright 2011 American Mathematical Society

American Mathematical Society