2015/05/19 by Alex C. Dantas, Dantas, Alex C., Said N. Sidki +1
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR
paper · pdf · doi:10.48550/arxiv.1505.05165
arxiv created 2015/05/19 · openalex publication_date 2015/05/19 · arxiv updated 2015/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.