Asymptotic cones and Assouad-Nagata dimension
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- by J. Dydak and J. Higes PDF
- Proc. Amer. Math. Soc. 136 (2008), 2225-2233 Request permission
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
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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
- 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
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2383529