Assouad–Nagata dimension of locally finite groups and asymptotic cones

AbstractIn 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