On the lattice of subgroups of the lamplighter group

R. Grigorchuk, R. Kravchenko

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

Abstract

The techniques of modules and actions of groups on rooted trees are applied to study the subgroup structure and the lattice subgroup of lamplighter type groups of the form ζn,p = (Z/pZ)n Z for n ≥ 1 and p prime. We completely characterize scale invariant structures on ζ1,2. We determine all points on the boundary of binary tree (on which ζ1,p naturally acts in a self-similar manner) with trivial stabilizer. We prove the congruence subgroup property (CSP) and as a consequence show that the profinite completion ζ1,p of ζ1,p is a self-similar group generated by finite automaton. We also describe the structure of portraits of elements of ζ1,p and ζ1,p and show that ζ1,p is not a sofic tree shift group in the terminology of [T. Ceccherini-Silberstein, M. Coornaert, F. Fiorenza and Z. Sunic, Cellular automata between sofic tree shifts, Theor. Comput. Sci.506 (2013) 79-101; A. Penland and Z. Sunic, Sofic tree shifts and self-similar groups, preprint].

Original languageEnglish (US)
Pages (from-to)837-877
Number of pages41
JournalInternational Journal of Algebra and Computation
Volume24
Issue number6
DOIs
StatePublished - 2014

Keywords

  • Lamplighter group
  • Mealey automaton
  • automaton group
  • congruence subgroup property
  • profinite group
  • rank gradient
  • scale invariant group
  • self-similar group
  • subgroup growth

ASJC Scopus subject areas

  • Mathematics(all)

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