On Asymptotic Assouad-Nagata Dimension

2006/07/06 by Dranishnikov, A. N., Smith, J. · 2 citations
#51F99 #54F45 #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.math/0607143

Abstract

For a large class of metric space X including discrete groups we prove that the asymptotic Assouad-Nagata dimension AN-asdim X of X coincides with the covering dimension dim(ν_L X) of the Higson corona of X with respect to the sublinear coarse structure on X. Then we apply this fact to prove the equality AN-asdim(X x R) = AN-asdim X + 1. We note that the similar equality for Gromov's asymptotic dimension asdim generally fails to hold [Dr3]. Additionally we construct an injective map from the asymptotic cone without the basepoint to the sublinear Higson Corona.

Cited by

• Nerve-type and invariance theorems for asymptotic dimension
• Isoperimetric inequalities in Hadamard spaces of asymptotic rank two

Related

• Nagata-Assouad dimension via Lipschitz extensions
• Assouad-Nagata dimension of tree-graded spaces
• On asymptotic dimension and a property of Nagata
• Assouad-Nagata dimension of locally finite groups and asymptotic cones
• Large scale absolute extensors