Let $G_{ p,d}$ be the restricted wreath product $C_{ p} wr C^{ d}$ where $C_{ p}$ is a cyclic group of order a prime $p$ and $C^{ d}$ a free abelian group of finite rank $d$. We study the existence of faithful state-closed (fsc) representations of $G_{ p,d}$ on the $1$-rooted $m$-ary tree for some finite $m$. The group $G_{ 2,1}$, known as the lamplighter group, admits an fsc representation on the binary tree. We prove that for $d \geq 2$ there are no fsc representations of $G_{ p,d}$ on the $p$ -adic tree. We characterize all fsc representations of $G=G_{ p,1}$ on the $p$-adic tree where the first level stabilizer of the image of $G$ contains its commutator subgroup. Furthermore, for $d \geq 2$, we construct uniformly fsc representations of $G_{ p,d}$ on the $p^{ 2}$ -adic tree and exhibit concretely the representation of $G_{ 2,2}$ on the $4$-tree as a finite-state automaton group.
Categories: math.GR