Assouad-Nagata dimension of wreath products of groups

Abstract

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 ${\dim }_{AN}(H\wr G)={\dim }_{AN}(G)\le 1$.

11 pages, new references added