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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 offinite 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 fscrepresentation on the binary tree. We prove that for $d \geq 2$ there are nofsc representations of $G_{p,d}$ on the $p$ -adic tree. We characterize all fscrepresentations of $G=G_{p,1}$ on the $p$-adic tree where the first levelstabilizer 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 $G_{2,2}$ onthe $4$-tree as a finite-state automaton group.

On state-closed representations of restricted wreath product of groups of type G_{p,d}=C_{p}wrC^{d}