Asymptotic cones and Assouad-Nagata dimension

Details Asymptotic cones and Assouad-Nagata dimension Asymptotic cones and Assouad-Nagata dimension

2006-10-10

arxiv.org/abs/math/0610338v3

Proceedings of the American Math.Soc. 136 (2008), 2225--2233

Mathematics

Metric Geometry

11 pages, comments from Behrstock, Brodskiy, Dranishnikov included

Scientific paper

We prove the dimension of any asymptotic cone over a metric space X does not exceed the asymptotic Assouad-Nagata dimension of X. This improves a result of Dranishnikov and Smith who showed that dim(Y) does not exceed asymptotic Assouad-Nagata dimension of X for all separable subsets Y of special asymptotic cones of X over an exponential ultrafilter. We also show that Assouad-Nagata dimension of the discrete Heisenberg group equals its asymptotic dimension.

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