  Consider the wreath product $H\wr G$, where $H\ne 1$ is finite and $G$ is
finitely generated. We show that the Assouad-Nagata dimension $\dim_{AN}(H\wr
G)$ of $H\wr G$ depends on the growth of $G$ as follows: If the growth of $G$
is not bounded by a linear function, then $\dim_{AN}(H\wr G)=\infty$, otherwise
$\dim_{AN}(H\wr G)=\dim_{AN}(G)\leq 1$.
