Assouad-Nagata dimension of locally finite groups and asymptotic cones

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

15 pages, added proposition 3.2, corrected misprints and improved some proofs

References in corpus (2)

• Hurewicz Theorem for Assouad-Nagata dimension
• Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions

Cited by in corpus (1)

• The coarse classification of countable abelian groups