Asymptotic cones and Assouad-Nagata dimension

Dydak, J.; Higes, J.

doi:10.1090/S0002-9939-08-09149-1

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ISSN 1088-6826(online) ISSN 0002-9939(print)

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
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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