On state-closed representations of restricted wreath product of groups of type Gₚ,d=CₚwrCᵈ

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Let Gₚ,d be the restricted wreath product Cₚ wr Cᵈ where Cₚ is a cyclic group of order a prime p and Cᵈ a free abelian group of finite rank d. We study the existence of faithful state-closed (fsc) representations of Gₚ,d on the 1-rooted m-ary tree for some finite m. The group G₂,₁, known as the lamplighter group, admits an fsc representation on the binary tree. We prove that for d ≥ 2 there are no fsc representations of Gₚ,d on the p -adic tree. We characterize all fsc representations of G=Gₚ,₁ on the p-adic tree where the first level stabilizer of the image of G contains its commutator subgroup. Furthermore, for d ≥ 2, we construct uniformly fsc representations of Gₚ,d on the p² -adic tree and exhibit concretely the representation of G₂,₂ on the 4-tree as a finite-state automaton group.