scholarly journals The profinite completion of multi-EGS groups

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Anitha Thillaisundaram ◽  
Jone Uria-Albizuri
Keyword(s):  
Congruence Subgroup ◽  
Profinite Completion ◽  
Congruence Subgroup Property

AbstractThe class of multi-EGS groups is a generalisation of the well-known Grigorchuk–Gupta–Sidki (GGS-)groups. Here we classify branch multi-EGS groups with the congruence subgroup property and determine the profinite completion of all branch multi-EGS groups. Additionally, our results show that branch multi-EGS groups are just infinite.

scholarly journals On the lattice of subgroups of the lamplighter group

2014 ◽  
Vol 24 (06) ◽  
pp. 837-877 ◽  
Author(s):  
R. Grigorchuk ◽  
R. Kravchenko
Keyword(s):  
Binary Tree ◽  
Congruence Subgroup ◽  
Profinite Completion ◽  
Scale Invariant ◽  
Lamplighter Group ◽  
Invariant Structures ◽  
Self Similar ◽  
Lattice Subgroup ◽  
Lattice Of Subgroups ◽  
Congruence Subgroup Property

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 ℒn,p = (ℤ/pℤ)n ≀ ℤ for n ≥ 1 and p prime. We completely characterize scale invariant structures on ℒ1,2. We determine all points on the boundary of binary tree (on which ℒ1,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 [Formula: see text] of ℒ1,p is a self-similar group generated by finite automaton. We also describe the structure of portraits of elements of ℒ1,p and [Formula: see text] and show that ℒ1,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].

scholarly journals Profinite invariants of arithmetic groups

2020 ◽  
Vol 8 ◽  
Author(s):  
Holger Kammeyer ◽  
Steffen Kionke ◽  
Jean Raimbault ◽  
Roman Sauer
Keyword(s):  
Euler Characteristic ◽  
Finite Groups ◽  
Congruence Subgroup ◽  
Arithmetic Groups ◽  
Profinite Completion ◽  
Residually Finite ◽  
Residually Finite Groups ◽  
Congruence Subgroup Property

Abstract We prove that the sign of the Euler characteristic of arithmetic groups with the congruence subgroup property is determined by the profinite completion. In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of type F. Our methods imply similar results for $\ell^2$ -torsion as well as a strong profiniteness statement for Novikov–Shubin invariants.

A PROBABILISTIC ZETA FUNCTION FOR ARITHMETIC GROUPS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 1053-1059 ◽  
Author(s):  
AVINOAM MANN
Keyword(s):  
Analytic Function ◽  
Zeta Function ◽  
Half Plane ◽  
Congruence Subgroup ◽  
Profinite Group ◽  
Arithmetic Groups ◽  
Finitely Generated ◽  
Profinite Completion ◽  
Random Elements ◽  
Congruence Subgroup Property

A profinite group G is positively finitely generated (PFG) if for some k, the probability P(G,k) that k random elements generate G is positive. It was conjectured that if G is PFG, then the function P(G,k) can be interpolated to an analytic function defined in some right half-plane. Here that conjecture is formulated more precisely, and verified for (the profinite completion of) arithmetic groups satisfying the congruence subgroup property.

Valuation-like maps and the congruence subgroup property

2001 ◽  
Vol 144 (3) ◽  
pp. 571-607 ◽  
Author(s):  
Andrei S. Rapinchuk ◽  
Yoav Segev
Keyword(s):  
Congruence Subgroup ◽  
Congruence Subgroup Property

scholarly journals On the congruence subgroup property for GGS-groups

2017 ◽  
Vol 145 (8) ◽  
pp. 3311-3322 ◽  
Author(s):  
Gustavo A. Fernández-Alcober ◽  
Alejandra Garrido ◽  
Jone Uria-Albizuri
Keyword(s):  
Congruence Subgroup ◽  
Congruence Subgroup Property

scholarly journals Multi-GGS Groups have the Congruence Subgroup Property

2019 ◽  
Vol 62 (3) ◽  
pp. 889-894 ◽  
Author(s):  
Alejandra Garrido ◽  
Jone Uria–Albizuri
Keyword(s):  
Congruence Subgroup ◽  
Constant Vector ◽  
General Proof ◽  
The Family ◽  
The One ◽  
Congruence Subgroup Property

AbstractWe generalize the result about the congruence subgroup property for GGS groups in [3] to the family of multi-GGS groups; that is, all multi-GGS groups except the one defined by the constant vector have the congruence subgroup property. New arguments are provided to produce this more general proof.

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