Asymptotic cones and Assouad-Nagata dimension
Copyright policy )Asymptotic cones and Assouad-Nagata dimension
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.
- Complutense University of Madrid Spain
- University of Tennessee at Knoxville United States
Mathematics - Geometric Topology, Mathematics - Metric Geometry, FOS: Mathematics, Metric Geometry (math.MG), Geometric Topology (math.GT)
Mathematics - Geometric Topology, Mathematics - Metric Geometry, FOS: Mathematics, Metric Geometry (math.MG), Geometric Topology (math.GT)
citations This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).12 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Average influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Top 10% impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average
Citations provided by BIP!Popularity provided by BIP!