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ISSN 1088-6826(e) ISSN 0002-9939(p)

Asymptotic cones and Assouad-Nagata dimension

Author(s): J. Dydak; J. Higes
Journal: Proc. Amer. Math. Soc. 136 (2008), 2225-2233.
MSC (2000): Primary 54F45; Secondary 55M10, 54C65
Posted: 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 . This improves a result of Dranishnikov and Smith (2007), who showed $\dim(Y)\leq \operatorname{asdim}_{AN}(X)$ for all separable subsets 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: 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
Posted: 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
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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