Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the structure of finitely generated subgroups of branch groups

Dominik Francoeur, Rostislav Grigorchuk, Paul-Henry Leemann, Tatiana Nagnibeda

TL;DR

This work develops a structural theory for finitely generated subgroups of branch groups satisfying the subgroup induction property (SIP), with a focus on tree-primitive actions. By introducing block subgroups, diagonal blocks, full supporting sets, and the dependence function, the authors prove that every finitely generated subgroup is virtually a block subgroup, and under additional hypotheses, virtually a regular block subgroup over the maximal branching subgroup. The results apply to key examples such as the Grigorchuk group $\mathcal{G}$ and torsion GGS groups, yielding precise block decompositions and implications for profinite-closure properties (LERF). The paper thus provides a comprehensive framework connecting SIP, tree-primitivity, and block-structure descriptions, advancing understanding of subgroup lattices in self-similar branch groups and their LERF/CSP-type properties.

Abstract

We describe the block structure of finitely generated subgroups of branch groups with the so-called subgroup induction property, including the first Grigorchuk group $\mathcal{G}$ and the torsion GGS groups.

On the structure of finitely generated subgroups of branch groups

TL;DR

This work develops a structural theory for finitely generated subgroups of branch groups satisfying the subgroup induction property (SIP), with a focus on tree-primitive actions. By introducing block subgroups, diagonal blocks, full supporting sets, and the dependence function, the authors prove that every finitely generated subgroup is virtually a block subgroup, and under additional hypotheses, virtually a regular block subgroup over the maximal branching subgroup. The results apply to key examples such as the Grigorchuk group and torsion GGS groups, yielding precise block decompositions and implications for profinite-closure properties (LERF). The paper thus provides a comprehensive framework connecting SIP, tree-primitivity, and block-structure descriptions, advancing understanding of subgroup lattices in self-similar branch groups and their LERF/CSP-type properties.

Abstract

We describe the block structure of finitely generated subgroups of branch groups with the so-called subgroup induction property, including the first Grigorchuk group and the torsion GGS groups.
Paper Structure (15 sections, 33 theorems, 39 equations, 3 figures)

This paper contains 15 sections, 33 theorems, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on branch groups
  3. The poset of words over a finite alphabet and branch groups
  4. Self-similar groups and regular branch groups
  5. Normal and almost normal subgroups of branch groups
  6. Normal subgroups of branch groups
  7. Almost normal subgroups of branch groups
  8. Tree-primitive actions
  9. Block subgroups and the subgroup induction property
  10. Block subgroups
  11. The subgroup induction property
  12. Block subgroups of branch groups with the SIP
  13. Full supporting sets and the dependence function
  14. Block subgroups in groups with the SIP
  15. Regular block subgroups of groups with the SIP

Key Result

Theorem 1.1

Let $G$ be a finitely generated self-replicating branch group with the subgroup induction property acting tree-primitively on a regular rooted tree, and let $H\leq G$ be a finitely generated subgroup of $G$. Then, $H$ is virtually a block subgroup.

Figures (3)

  • Figure 1: A block subgroup of a self-similar group $G$. Here $B$ and $K$ are finite index subgroups of $G$.
  • Figure 2: A diagonal block subgroup over $K$ with supporting vertex set of cardinality $3$.
  • Figure 3: A block subgroup with supporting partition $\{\{000,001\},1\}$. Here $H=H_1\cdot H_2$ with $H_1$ a block subgroup over $K$ with supporting partition $\{000,001\}$ and $H_2$ a block subgroup over $B$ with supporting partition $\{B\}$.

Theorems & Definitions (98)

  • Theorem 1.1: Theorem \ref{['thm:BlockSubgroups']}
  • Theorem 1.2: Theorem \ref{['thm:RegularBlockSubgroup']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Example 2.8: The Grigorchuk group
  • ...and 88 more