Lower central series of a group of tree automorphisms
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 \(n\).
scientific article; zbMATH DE number 1158888