Asymptotic cones and Assouad-Nagata dimension

Dydak, J.; Higes, J.

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

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.

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