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
SN - 0218-1967
VL - 24
SP - 837
EP - 877
JO - International Journal of Algebra and Computation
JF - International Journal of Algebra and Computation
IS - 6
ER -