Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Asymptotic cones and Assouad-Nagata dimension


Authors: J. Dydak and J. Higes
Journal: Proc. Amer. Math. Soc. 136 (2008), 2225-2233
MSC (2000): Primary 54F45; Secondary 55M10, 54C65
DOI: https://doi.org/10.1090/S0002-9939-08-09149-1
Published electronically: February 14, 2008
MathSciNet review: 2383529
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that the dimension of any asymptotic cone over a metric space $ (X,\rho)$ does not exceed the asymptotic Assouad-Nagata dimension $ \operatorname{asdim}_{AN}(X)$ of $ X$. This improves a result of Dranishnikov and Smith (2007), who showed $ \dim(Y)\leq \operatorname{asdim}_{AN}(X)$ for all separable subsets $ Y$ of special asymptotic cones $ \operatorname{Cone}_\omega(X)$, where $ \omega$ is an exponential ultrafilter on natural numbers.

We also show that the Assouad-Nagata dimension of the discrete Heisenberg group equals its asymptotic dimension.


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

  • 1. P. Assouad, Sur la distance de Nagata, C. R. Acad. Sci. Paris Ser. I Math. 294 (1982), no. 1, 31-34. MR 651069 (83b:54034)
  • 2. J. Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space, Geom. Topol. 10 (2006), 1523-1578. MR 2255505
  • 3. J. Behrstock and Y.N. Minsky, Dimension and rank for mapping class groups, preprint ArXiv:math.GT/0512352.
  • 4. G. Bell and A. Dranishnikov, A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory, preprint, math.GR/0407431 (2004).
  • 5. G. Bell and A. Dranishnikov, Asymptotic dimension in Bedlewo, Topology Proceedings (to appear).
  • 6. N. Brodskiy, J. Dydak, J. Higes, A. Mitra, Nagata-Assouad dimension via Lipschitz extensions, preprint ArXiv:math.MG/0601226, Israel Journal of Math. (to appear).
  • 7. N. Brodskiy, J. Dydak, M. Levin, A. Mitra, Hurewicz Theorem for Assouad-Nagata dimension, preprint ArXiv:math.MG/0605416, Journal of the London Math. Soc. (to appear).
  • 8. J. Burillo, Dimension and fundamental groups of asymptotic cones, Journal of the London Math. Soc. 59 (1999), 557-572. MR 1709665 (2000i:20067)
  • 9. A. Dranishnikov and J. Smith, Asymptotic dimension of discrete groups, Fund. Math. 189 (2006), 27-34. MR 2213160 (2007h:20041)
  • 10. A. N. Dranishnikov and J. Smith, On asymptotic Assouad-Nagata dimension, Topology Appl. 154 (2007), 934-952. MR 2294641
  • 11. A. Dranishnikov, M. Zarichnyi, Universal spaces for asymptotic dimension, Topology Appl. 140 (2004), nos. 2-3, 203-225. MR 2074917 (2005e:54032)
  • 12. C. Drutu, Quasi-isometry invariants and asymptotic cones, Int. J. Algebra Comput. 12 (1 and 2) (2002), 99-135. MR 1902363 (2003g:20069)
  • 13. C. Drutu, M. Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005), 959-1058. MR 2153979 (2006d:20078)
  • 14. M. Gromov, Groups of polynomial growth and expanding maps, Publ. Math. IHES 53 (1981), 53-73. MR 623534 (83b:53041)
  • 15. M. Gromov, Asymptotic invariants for infinite groups, in Geometric Group Theory, vol. 2, 1-295, G. Niblo and M. Roller, eds., Cambridge University Press, 1993. MR 1253544 (95m:20041)
  • 16. J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917 (2002c:30028)
  • 17. U. Lang, T. Schlichenmaier, Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions, IMRN International Mathematics Research Notices (2005), no. 58, 3625-3655. MR 2200122 (2006m:53061)
  • 18. M. Kapovich, Lectures on Geometric Group Theory, preprint (as of September 28, 2005).
  • 19. J. Nagata, Note on dimension theory of metric spaces, Fund. Math. 45 (1958), 143-181. MR 0105081 (21:3827)
  • 20. P. W. Nowak, On exactness and isoperimetric profiles of discrete groups, J. Funct. Anal. 243 (2007), 323-344. MR 2291440
  • 21. D. V. Osin, Subgroup distortions in nilpotent groups, Comm. Algebra 29 (2001), 5439-5463. MR 1872804 (2002j:20078)
  • 22. P. Pansu, Croissance des boules et des géodésiques fermées dans les nilvariétés, Ergod. Th. Dynam. Syst. 3 (1983), 415-445. MR 741395 (85m:53040)
  • 23. J. Roe, Lectures on coarse geometry, University Lecture Series, 31, American Mathematical Society, Providence, RI, 2003. MR 2007488 (2004g:53050)
  • 24. L. van den Dries, A. J. Wilkie, Gromov's theorem on groups of polynomial growth and elementary logic, J. Algebra 89 (1984), 349-374. MR 751150 (85k:20101)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 54F45, 55M10, 54C65

Retrieve articles in all journals with MSC (2000): 54F45, 55M10, 54C65


Additional Information

J. Dydak
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996
Email: dydak@math.utk.edu

J. Higes
Affiliation: Departamento de Geometría y Topología, Facultad de CC.Matemáticas, Universidad Complutense de Madrid, Madrid, 28040 Spain
Email: josemhiges@yahoo.es

DOI: https://doi.org/10.1090/S0002-9939-08-09149-1
Keywords: Assouad-Nagata dimension, asymptotic dimension, asymptotic cones, covering dimension
Received by editor(s): October 20, 2006
Published electronically: February 14, 2008
Additional Notes: The first author was partially supported by the Center for Advanced Studies in Mathematics at Ben Gurion University of the Negev (Beer-Sheva, Israel)
The second author is supported by Grant AP2004-2494 from the Ministerio de Educación y Ciencia, Spain
Communicated by: Alexander N. Dranishnikov
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society