  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
