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Subgroup induction property for branch groups

Dominik Francoeur, Paul-Henry Leemann

TL;DR

The paper investigates the subgroup induction property (SIP) for branch groups, distinguishing a weaker SIP from the full SIP and showing that SIP has strong consequences for structure and separability of subgroups. It proves that finitely generated branch groups with the weak SIP are torsion and just infinite, with further implications on maximal subgroups and commensurability of subgroups under self-replication. The second main contribution is proving that all torsion GGS groups possess the SIP, thereby providing an infinite family of SIP groups and yielding subgroup separability for these groups. The results broaden understanding of the subgroup structure in self-similar branch groups and have downstream effects on membership problems, Cantor–Bendixson ranks, and cohomology in this domain. Overall, the work significantly expands the catalog of groups with SIP and clarifies when weak SIP suffices or strengthens to SIP, with notable impact on GGS groups and their subgroups.

Abstract

The subgroup induction property is a property of self-similar groups acting on rooted trees introduced by Grigorchuk and Wilson in 2003 that appears to have strong implications on the structure of the groups possessing it. It was for example used in the proof that the first Grigorchuk group as well as the Gupta-Sidki 3-group are subgroup separable (locally extended residually finite) or to describe their finitely generated subgroups as well as their weakly maximal subgroups. However, until now, there were only two known examples of groups with this property, namely the first Grigorchuk group and the Gupta-Sidki 3-group. The aim of this article is twofold. First, we investigate various consequences of the subgroup induction property for branch groups, a particularly interesting class of self-similar groups. Notably, we show that finitely generated branch groups with the subgroup induction property must be torsion, just infinite and subgroup separable, and we establish conditions under which all their maximal subgroups are of finite index and all their weakly maximal subgroups are closed in the profinite topology. Then, we show that every torsion GGS group has the subgroup induction property, hence providing the first infinite family of examples of groups with this property.

Subgroup induction property for branch groups

TL;DR

The paper investigates the subgroup induction property (SIP) for branch groups, distinguishing a weaker SIP from the full SIP and showing that SIP has strong consequences for structure and separability of subgroups. It proves that finitely generated branch groups with the weak SIP are torsion and just infinite, with further implications on maximal subgroups and commensurability of subgroups under self-replication. The second main contribution is proving that all torsion GGS groups possess the SIP, thereby providing an infinite family of SIP groups and yielding subgroup separability for these groups. The results broaden understanding of the subgroup structure in self-similar branch groups and have downstream effects on membership problems, Cantor–Bendixson ranks, and cohomology in this domain. Overall, the work significantly expands the catalog of groups with SIP and clarifies when weak SIP suffices or strengthens to SIP, with notable impact on GGS groups and their subgroups.

Abstract

The subgroup induction property is a property of self-similar groups acting on rooted trees introduced by Grigorchuk and Wilson in 2003 that appears to have strong implications on the structure of the groups possessing it. It was for example used in the proof that the first Grigorchuk group as well as the Gupta-Sidki 3-group are subgroup separable (locally extended residually finite) or to describe their finitely generated subgroups as well as their weakly maximal subgroups. However, until now, there were only two known examples of groups with this property, namely the first Grigorchuk group and the Gupta-Sidki 3-group. The aim of this article is twofold. First, we investigate various consequences of the subgroup induction property for branch groups, a particularly interesting class of self-similar groups. Notably, we show that finitely generated branch groups with the subgroup induction property must be torsion, just infinite and subgroup separable, and we establish conditions under which all their maximal subgroups are of finite index and all their weakly maximal subgroups are closed in the profinite topology. Then, we show that every torsion GGS group has the subgroup induction property, hence providing the first infinite family of examples of groups with this property.

Paper Structure

This paper contains 13 sections, 42 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Organisation
  3. Acknowledgements
  4. Reminders on branch groups
  5. The subgroup induction property
  6. Definitions and preleminaries
  7. Consequences of the weak subgroup induction property
  8. Consequences of the subgroup induction property
  9. Weak subgroup induction property versus subgroup induction property.
  10. Subgroup induction property for GGS groups
  11. General facts about GGS groups
  12. Lengths in GGS groups
  13. Proof of the subgroup induction property for torsion GGS groups

Key Result

Theorem A

Let $G$ be a finitely generated branch group with the weak subgroup induction property. Then $G$ is torsion and just infinite. If moreover the intersection of all finite-index maximal subgroups of $G$ is not trivial (e.g. if $G$ is a $p$-group), then every maximal subgroup of $G$ is of finite index.

Theorems & Definitions (84)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 3.3
  • ...and 74 more