Assouad-Nagata dimension of locally finite groups and asymptotic cones |
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Authors: | J. Higes |
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Affiliation: | Departamento de Geometría y Topología, Facultad de Cc. Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain |
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Abstract: | 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 dG. If dimAN(G,dG)>0, is dimAN(G,dG) 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. |
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Keywords: | Assouad-Nagata dimension Asymptotic dimension Asymptotic cones Locally finite groups Ultrametric spaces |
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