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Asymptotic cones and Assouad-Nagata dimension

Authors:	J Dydak J Higes

Institution:	Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996 ; Departamento de Geometría y Topología, Facultad de CC.Matemáticas, Universidad Complutense de Madrid, Madrid, 28040 Spain

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.

Keywords:	Assouad-Nagata dimension asymptotic dimension asymptotic cones covering dimension

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