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On asymptotic Assouad–Nagata dimension
Journal article   Open access   Peer reviewed

On asymptotic Assouad–Nagata dimension

A.N. Dranishnikov and J. Smith
Topology and its applications, v 154(4), pp 934-952
15 Feb 2007
DOI: https://doi.org/10.1016/j.topol.2006.10.010
url
https://doi.org/10.1016/j.topol.2006.10.010View
Published, Version of Record (VoR)Open Access (Publisher-Specific),  Open

Abstract

Assouad–Nagata dimension Asymptotic dimension Dimension Higson corona
For a large class of metric spaces X including discrete groups we prove that the asymptotic Assouad–Nagata dimension AN-asdim X of X coincides with the covering dimension dim ( ν L X ) of the Higson corona of X with respect to the sublinear coarse structure on X. Then we apply this fact to prove the equality AN-asdim ( X × R ) = AN-asdim X + 1 . We note that the similar equality for Gromov's asymptotic dimension asdim generally fails to hold [A. Dranishnikov, Cohomological approach to asymptotic dimension, Preprint, 2006]. Additionally we construct an injective map ξ : cone ω ( X ) ∖ [ x 0 ] → ν L X from the asymptotic cone without the basepoint to the sublinear Higson corona.

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Mathematics
Mathematics, Applied
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