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 does not exceed the asymptotic Assouad-Nagata dimension
of
. This improves a result of Dranishnikov and Smith (2007), who showed
for all separable subsets
of special asymptotic cones
, where
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:
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.