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

Abstract

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 $d\ge 2$ 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 $d\ge 2$, 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.

References in corpus (1)

• The lamplighter group ${\mathbb{Z}}_3\wr \mathbb{Z}$ generated by a bireversible automaton