Assouad-Nagata dimension of locally finite groups and asymptotic cones

2007/11/09 by Higes, J.
#54C65 (Secondary) #54F45 (Primary) #55M10 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.0711.1512

Abstract

In this work we study two problems about Assouad-Nagata dimension: 1) Is there a metric space of non zero Assouad-Nagata dimension such that all of its asymptotic cones are of Assouad-Nagata dimension zero? (Dydak and Higes) 2) Suppose G is a locally finite group with a proper left invariant metric d_G. If dim_AN(G, d_G)>0, is dim_AN (G, d_G) infinite? (Brodskiy, Dydak and Lang) The first question is answered positively not only for general metric spaces but also for discrete groups with proper left invariant metrics. The second question has a negative solution. We show that for each n there exists a locally finite group of Assouad-Nagata dimension n. A generalization to countable groups of arbitrary asymptotic dimension is given

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