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

Let G(p,d) be the restricted wreath product C(p)wrC(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 transitive state-closed (fsc) representations of C-p,C-d on the rooted m-ary tree for some finite in. The group G(2,1), 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 G(p,d) on the p-adic tree. We describe all fsc representations of G = G(p,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 Cp,d on the p2-adic tree and exhibit concretely the representation of C-2,C-2 on the 4-tree as a finite-state automaton group. (C) 2017 Elsevier Inc. All rights reserved.