Asymptotic cones and Assouad-Nagata dimension

“…We will also give some examples of metric spaces of infinite asymptotic Assouad-Nagata dimension such that all of its asymptotic cones are ultrametric. These two results will solve Question 4.5 of [11].…”

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confidence: 79%

Assouad–Nagata dimension of locally finite groups and asymptotic cones

Higes

2010

Topology and its Applications

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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]).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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“…In [15] the authors show the following relationship between the topological dimension of an asymptotic cone and the asymptotic Assouad-Nagata dimension of the space:…”

Section: Lower Bounds For Assouad-nagata Dimensionmentioning

confidence: 99%

“…That dimension can be considered as the linear version of the asymptotic dimension. In recent years a part of the research activity was focused on this dimension and its relationship with the asymptotic dimension (see for example [19], [12], [13], [6], [7], [9], [8], [20], [15], [18]). One of the main problems of interest consists of studying the differences between the asymptotic dimension and the asymptotic Assouad-Nagata dimension in the context of the geometric group theory.…”

Section: Introductionmentioning

confidence: 99%