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Maximal subgroups in torsion branch groups

Mikel Eguzki Garciarena, J. Moritz Petschick

Abstract

We study the maximal subgroups of branch groups and obtain a criterion that ensures that certain spinal groups are contained in the class $\mathcal{MF}$ of groups with all maximal subgroups of finite index. This allows us to construct branch groups within $\mathcal{MF}$ exhibiting novel properties, for example groups that possess non-normal maximal subgroups. Furthermore, we give new concrete examples of branch groups outside $\mathcal{MF}$, with explicitly given maximal subgroups of infinite index. Most prominently, we construct a periodic branch group outside $\mathcal{MF}$.

Maximal subgroups in torsion branch groups

Abstract

We study the maximal subgroups of branch groups and obtain a criterion that ensures that certain spinal groups are contained in the class of groups with all maximal subgroups of finite index. This allows us to construct branch groups within exhibiting novel properties, for example groups that possess non-normal maximal subgroups. Furthermore, we give new concrete examples of branch groups outside , with explicitly given maximal subgroups of infinite index. Most prominently, we construct a periodic branch group outside .

Paper Structure

This paper contains 19 sections, 47 theorems, 144 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Rooted trees and constant spinal groups
  3. Notation
  4. Graphs
  5. Rooted trees and their automorphisms
  6. Finite groups as alphabets
  7. Groups of automorphisms
  8. Maximal subgroups of branch groups
  9. Constant spinal groups
  10. Constant spinal groups with all maximal subgroups of finite index
  11. General strategy to establish membership in $\mathcal{MF}$
  12. Obtaining a directed element
  13. Filling directed elements
  14. Delta and theta maps
  15. Contraction under the maps $\theta_d$
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $G = \langle R \cup D \rangle$ be a periodic primary multi-GGS group acting on the rooted tree $\underline{R}^\ast$. Assume that there exists a directed element $d \in D$ and a vertex $x \in \underline{R}$ corresponding to a generator of $R$, and such that $d|_{x}$ also generates $R$. Then $G$ i

Figures (3)

  • Figure 1: The graph $\mathcal{R}^{\mathrm{ab}}(b)$ associated to the (primary) GGS group $\langle \{a, b\} \rangle$ with defining vector $(1, 2, 2) \in (\mathop{\mathrm{\mathbb{Z}}}\nolimits/4\mathop{\mathrm{\mathbb{Z}}}\nolimits)^3$ (left) and $(1, 1, 0) \in (\mathop{\mathrm{\mathbb{Z}}}\nolimits/4\mathop{\mathrm{\mathbb{Z}}}\nolimits)^3$ (right). The only forking point is the vertex $a$ in the second graph, whose reach is $\{\mathop{\mathrm{id}}\nolimits, a\}$ and whose reach closure is the full rooted group.
  • Figure 2: The graph $\mathcal{R}^{\mathrm{ab}}(d^j)$ as described in the proof of \ref{['thm:non mn']}.
  • Figure 3: Parts of the graph $\mathcal{R}^{\mathrm{ab}}(d^j)$ for $j = 1$ to $4$ top left to bottom right as described in the proof of \ref{['thm:mf not just-insol']}. Not all edges and vertices are drawn. Unlabelled edges are thought to be labelled with $1$.

Theorems & Definitions (91)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 81 more