Asymptotic cones and Assouad-Nagata dimension

Abstract

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.