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

Let Gp,d be the restricted wreath product CpwrCd 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 transitive state-closed (fsc) representations of Gp,d on the 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 describe all fsc representations of G=Gp,1 on the p-adic tree obtained via virtual endomorphisms, where the first level stabilizer of the image of G contains its commutator subgroup. Furthermore, for d≥2, we construct 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.