Projects per year
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.
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.
Original language | English |
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Pages (from-to) | 175 |
Number of pages | 207 |
Journal | Journal of Fractal Geometry |
Volume | 12 |
Issue number | 1/2 |
DOIs | |
Publication status | Published - 16 Jan 2025 |
Keywords
- branch groups
- finitely generated subgroups
- just infinite groups
- maximal subgroups
- LERF
- subgroup separability
Projects
- 0 Active
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Groups acting on rooted trees and their subgroups
1/01/24 → 31/12/26
Project: Internal Research Project
Research output
- 0 Article
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Finitely generated subgroups of branch groups and subdirect products of just infinite groups
Grigorchuk, R. I., Leemann, P. H. & Nagnibeda, T. V., Nov 2021, In: Izvestiya Mathematics. 85, 6, p. 1128-1145 18 p.Research output: Contribution to journal › Article › peer-review
Open Access
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Sous-groupes maximaux, faiblement maximaux et généralisations
Paul-Henry Leemann (Speaker)
4 Feb 2025Activity: Talk or presentation › Presentation at conference/workshop/seminar
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Nouveaux exemples de groupes séparables par les sous-groupes (LERF)
Paul-Henry Leemann (Speaker)
25 Jun 2024Activity: Talk or presentation › Invited talk
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University of Geneva
Paul-Henry Leemann (Visiting researcher)
10 Jun 2024 → 20 Jun 2024Activity: Research visit
Cite this
Leemann, P.-H., & Francoeur, D. (2025). Subgroup induction property for branch groups. Journal of Fractal Geometry, 12(1/2), 175. https://doi.org/10.4171/JFG/162