TY - JOUR

T1 - On the lattice of subgroups of the lamplighter group

AU - Grigorchuk, R.

AU - Kravchenko, R.

N1 - Funding Information:
We would like to express our thanks to Yves de Cornulier, Murray Elder, Mark Sapir and Dan Segal. We are indebted to Anna Erschler and the Laboratoire de Mathématiques d’Orsay for their hospitality. The major part of the work was done while the first author was visiting and the second author was a postdoc there. We would also like to thank the Mittag-Leﬄer institute and Institute Henri Poincare trimester program, where the final stages of the work were completed. The first author was partially supported by Simons Foundation and NSF grant DMS-1207699 and first and second authors were supported by Mittag-Leﬄer Institute and ERC starting grant GA 257110 RaWG.
Publisher Copyright:
© 2014 World Scientific Publishing Company.

PY - 2014

Y1 - 2014

N2 - 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].

AB - 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].

KW - Lamplighter group

KW - Mealey automaton

KW - automaton group

KW - congruence subgroup property

KW - profinite group

KW - rank gradient

KW - scale invariant group

KW - self-similar group

KW - subgroup growth

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U2 - 10.1142/S0218196714500374

DO - 10.1142/S0218196714500374

M3 - Article

AN - SCOPUS:84928540260

VL - 24

SP - 837

EP - 877

JO - International Journal of Algebra and Computation

JF - International Journal of Algebra and Computation

SN - 0218-1967

IS - 6

ER -