  We prove that the asymptotic Assouad-Nagata dimension of a connected Lie
group $G$ equipped with a left-invariant Riemannian metric coincides with its
topological dimension of $G/C$ where $C$ is a maximal compact subgroup. To
prove it we will compute the Assouad-Nagata dimension of connected solvable Lie
groups and semisimple Lie groups. As a consequence we show that the asymptotic
Assouad-Nagata dimension of a polycyclic group equipped with a word metric is
equal to its Hirsch length and that some wreath-type finitely generated groups
can not be quasi-isometric to any cocompact lattice on a connected Lie group.
