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On the lower central series of a large family of non-periodic GGS-groups

Gustavo A. Fernández-Alcober, Mikel E. Garciarena, Marialaura Noce

Abstract

For an odd prime $p$, we determine the lower central series of a large family of non-periodic GGS-groups, which has a density of roughly $(\frac{p-1}{p})^2$ within all GGS-groups. This means a significant extension of the knowledge regarding the lower central series of distinguished classes of branch groups, which to date was basically restricted to the Grigorchuk group. As part of our results, we obtain the indices between consecutive terms of the lower central series, and we show that these groups, as well as their profinite completions, have lower central width equal to $2$. In particular, this confirms a conjecture of Bartholdi, Eick, and Hartung about the generalised Fabrykowski-Gupta groups.

On the lower central series of a large family of non-periodic GGS-groups

Abstract

For an odd prime , we determine the lower central series of a large family of non-periodic GGS-groups, which has a density of roughly within all GGS-groups. This means a significant extension of the knowledge regarding the lower central series of distinguished classes of branch groups, which to date was basically restricted to the Grigorchuk group. As part of our results, we obtain the indices between consecutive terms of the lower central series, and we show that these groups, as well as their profinite completions, have lower central width equal to . In particular, this confirms a conjecture of Bartholdi, Eick, and Hartung about the generalised Fabrykowski-Gupta groups.

Paper Structure

This paper contains 5 sections, 28 theorems, 203 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and uniseriality in non-periodic GGS-groups
  3. The lower central series of some wreath products
  4. Lower central series of $G_n$ for $n\le 3$
  5. Proof of Theorem A

Key Result

Theorem 2.2

Let $G$ be a non-periodic GGS-group. Then the following hold for every $n\in\mathbb{N}$: In particular, $G_n$ acts uniserially on $\mathop{\mathrm{St}}\nolimits_{G_n}(n-1)$, and the only subgroups of $\mathop{\mathrm{St}}\nolimits_{G_n}(n-1)$ that are normal in $G_n$ are those of the form Also if $h$ is a generator of $\mathop{\mathrm{St}}\nolimits_{G_n}(n-1)$ modulo $[\mathop{\mathrm{St}}\nolim

Figures (2)

  • Figure 1: Position of the relevant subgroups in the interval $[r(n-2),c(n)]$.
  • Figure 2: Position of the relevant subgroups in the interval $[\ell(n-2),r(n-2)-1]$.

Theorems & Definitions (54)

  • Definition 1.1
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • Theorem 3.4
  • proof
  • ...and 44 more