Lower central series of a group of tree automorphisms
type: Item
instance of: scholarly article
title: Lower central series of a group of tree automorphisms
zbMATH Open document ID: 0899.20014
DOI: 10.1007/BF02305180
author: A. V. Rozhkov
published in: Mathematical Notes
publication date: 1998-09-15T00:00:00Z
review text: The Grigorchuk 3-generated 2-group \(H\) [\textit{R. I. Grigorchuk}, Funkts. Anal. Prilozh. 14, No. 1, 53-54 (1980; Zbl 0595.20029)] has lately been an object of intensive study. This group usually serves as the special test for many problems connected to Aleshin type groups. Therefore, any information about this group is very important. R. I. Grigorchuk proved the intermediate growth of this group, I. G. Lysenko described its defining relations, also its finite, abelian, normal subgroups, its centralizers and iterated centralizers of elements, its commutator series has been described [see for example, \textit{A. V. Rozhkov} Mat. Zametki 40, No. 5, 572-589 (1986; Zbl 0614.20019)]. In the article under review the author describes the lower central series of the Grigorchuk 3-generated 2-group and calculates the indexes of the factors. These factors are elementary abelian groups of rank 1 or 2. Also the nilpotent length and solvability length of the factors by congruence subgroups are determined. The following two theorems are main results of this article. Theorem 1. The following relations are true: \[ \begin{multlined} H/\gamma_2\cong\mathbb{Z}_2\oplus\mathbb{Z}_2\oplus\mathbb{Z}_2,\quad\gamma_2/\gamma_3\cong\mathbb{Z}_2\oplus\mathbb{Z}_2,\\ \gamma_n/\gamma_{n+1}\cong\begin{cases}\mathbb{Z}_2\oplus\mathbb{Z}_2 &\text{ if } 2^m+1\leq n<3\cdot 2^{m-1}+1,\\ \mathbb{Z}_2 &\text{ if } 3\cdot 2^{m-1}+1\leq n<2^{m+1}+1,\end{cases}\end{multlined} \] where \(m=1,2,\dots\). Furthermore, \(\gamma_r=N_m\), where \(r=2^m+1\), and \(N_{m+1}\leq\gamma_n\leq N_m\) for \(2^m+1\leq n\leq 2^{m+1}+1\), \(m=1,2,\dots\) Theorem 2. For any positive integer \(n\), \(n>3\), the nilpotent lengths of the factor groups \(H/st(n)\) and \(H/\text{cost}(n)\) are given by the functions \(2^{m-1}+1\) and \(2^{m+1}+1\), respectively. Their solvability lengths are equal to \((n+3)/2\) for \(n\) odd, to \((n+2)/2\) and \((n+4)/2\), respectively, for an even \(n\).
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20F50
zbMATH DE Number: 1158888
zbMATH Keywords: \(AT\)-groups
zbMATH Keywords: Aleshin type groups
zbMATH Keywords: intermediate growth
zbMATH Keywords: normal subgroups
zbMATH Keywords: centralizers of elements
zbMATH Keywords: lower central series
zbMATH Keywords: Grigorchuk 3-generated 2-group
zbMATH Keywords: nilpotent lengths
zbMATH Keywords: solvability lengths
reviewed by: Igor Subbotin
MaRDI profile type: Publication
cites work: On Burnside's problem on periodic groups
cites work: CENTRALIZERS OF ELEMENTS IN A GROUP OF TREE AUTOMORPHISMS
cites work: Q4722255
description: scientific article; zbMATH DE number 1158888