Assouad-Nagata dimension of locally finite groups and asymptotic cones

作者：Higes J^*

来源：Topology and Its Applications, 2010, 157(17): 2635-2645.

DOI：10.1016/j.topol.2010.07.015

• Assouad-Nagata dimension
• Asymptotic dimension
• Asymptotic cones
• Locally finite groups
• Ultrametric spaces

In this paper we study two problems concerning Assouad-Nagata dimension:
(1) Is there a metric space of positive asymptotic Assouad-Nagata dimension such that all of its asymptotic cones are of Assouad-Nagata dimension zero? (Dydak and Higes, 2008 [11, Question 4.5]).
(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 et al., preprint [6, Problem 5.3]).
The first question is answered positively. We provide examples of metric spaces of positive (even infinite) Assouad-Nagata dimension such that all of its asymptotic cones are ultrametric. The metric spaces can be 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. As a consequence this solves for non-finitely generated countable groups the question about the existence of metric spaces of finite asymptotic dimension whose asymptotic Assouad-Nagata dimension is larger but finite.

• 出版日期2010-11-1