Asymptotic cones and Assouad-Nagata dimension
J. Dydak, J. Higes
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Source: Crossref
Published: Feb 14, 2008
DOI: 10.1090/s0002-9939-08-09149-1
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We prove that the dimension of any asymptotic cone over a metric space ( X , ρ ) (X,\rho ) does not exceed the asymptotic Assouad-Nagata dimension asdim A N ( X ) \operatorname {asdim}_{AN}(X) of X X . This improves a result of Dranishnikov and Smith (2007), who showed dim ( Y ) ≤ asdim A N ( X ) \dim (Y)\leq \operatorname {asdim}_{AN}(X) for all separable subsets Y Y of special asymptotic cones Cone ω ( X ) \operatorname {Cone}_\omega (X) , where ω \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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