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Proceedings of the American Mathematical Society
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
Published electronically: February 14, 2008
MathSciNet review: 2383529
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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.


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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: http://dx.doi.org/10.1090/S0002-9939-08-09149-1
PII: S 0002-9939(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.