On the structure of finitely generated subgroups of branch groups

Dominik Francoeur; R. Grigorchuk; Paul-Henry Leemann; T. Nagnibeda

doi:10.4171/JCA/112

On the structure of finitely generated subgroups of branch groups

Dominik Francoeur
, R. Grigorchuk
, Paul-Henry Leemann^*
, T. Nagnibeda

^*Corresponding author for this work

Universidad Autónoma de Madrid
Texas A&M University
University of Geneva
St. Petersburg State University

Research output: Contribution to journal › Article › peer-review

Abstract

Motivated by the study of profinite topology in branch groups, we prove a structural result about their finitely generated subgroups. More precisely, we show that finitely generated subgroups of a branch group with the subgroup induction property have a block structure, which roughly means that, up to a finite index, they are products of finite index subgroups, embedded in the group in a way that is coherent with its branch action on the rooted tree.

Original language	English
Journal	Journal of Combinatorial Algebra
DOIs	https://doi.org/10.4171/JCA/112
Publication status	E-pub ahead of print - 14 May 2025

Keywords

Regular branch groups
self-replicating groups
block subgroup
finitely generated subgroups
subgroup induction property
tree-primitive action

Access to Document

10.4171/JCA/112

1 Active

Groups acting on rooted trees and their subgroups
Leemann, P.-H. (PI)
1/01/24 → 31/12/26
Project: Internal Research Project

3 Article

Subgroup induction property for branch groups
Leemann, P. H. & Francoeur, D., 16 Jan 2025, In: Journal of Fractal Geometry. 12, 1-2, p. 175-207 33 p.
Research output: Contribution to journal › Article › peer-review

Open Access
2 Citations (Scopus)
Weakly maximal subgroups of branch groups
Leemann, P. H., 30 May 2025, In: International Journal of Algebra and Computation. 35, 5, p. 587-646
Research output: Contribution to journal › Article › peer-review

Open Access
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

5 Presentation at conference/workshop/seminar
3 Research visit
1 Invited talk

On the structure of finitely generated subgroups of branch groups
Leemann, P.-H. (Speaker)
10 Jul 2025
Activity: Talk or presentation › Presentation at conference/workshop/seminar

File
Maximal subgroups, weakly maximal subgroups and generalization
Leemann, P.-H. (Speaker)
23 May 2025
Activity: Talk or presentation › Presentation at conference/workshop/seminar
Sous-groupes maximaux, faiblement maximaux et généralisations
Leemann, P.-H. (Speaker)
4 Feb 2025
Activity: Talk or presentation › Presentation at conference/workshop/seminar

Cite this

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title = "On the structure of finitely generated subgroups of branch groups",

abstract = "Motivated by the study of profinite topology in branch groups, we prove a structural result about their finitely generated subgroups. More precisely, we show that finitely generated subgroups of a branch group with the subgroup induction property have a block structure, which roughly means that, up to a finite index, they are products of finite index subgroups, embedded in the group in a way that is coherent with its branch action on the rooted tree.",

keywords = "Regular branch groups, self-replicating groups, block subgroup, finitely generated subgroups, subgroup induction property, tree-primitive action",

author = "Dominik Francoeur and R. Grigorchuk and Paul-Henry Leemann and T. Nagnibeda",

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doi = "10.4171/JCA/112",

language = "English",

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AU - Francoeur, Dominik

AU - Grigorchuk, R.

AU - Leemann, Paul-Henry

AU - Nagnibeda, T.

PY - 2025/5/14

Y1 - 2025/5/14

N2 - Motivated by the study of profinite topology in branch groups, we prove a structural result about their finitely generated subgroups. More precisely, we show that finitely generated subgroups of a branch group with the subgroup induction property have a block structure, which roughly means that, up to a finite index, they are products of finite index subgroups, embedded in the group in a way that is coherent with its branch action on the rooted tree.

AB - Motivated by the study of profinite topology in branch groups, we prove a structural result about their finitely generated subgroups. More precisely, we show that finitely generated subgroups of a branch group with the subgroup induction property have a block structure, which roughly means that, up to a finite index, they are products of finite index subgroups, embedded in the group in a way that is coherent with its branch action on the rooted tree.

KW - Regular branch groups

KW - self-replicating groups

KW - block subgroup

KW - finitely generated subgroups

KW - subgroup induction property

KW - tree-primitive action

U2 - 10.4171/JCA/112

DO - 10.4171/JCA/112

M3 - Article

SN - 2415-6302

JO - Journal of Combinatorial Algebra

JF - Journal of Combinatorial Algebra

ER -

On the structure of finitely generated subgroups of branch groups

Abstract

Keywords

Access to Document

Fingerprint

Projects

Groups acting on rooted trees and their subgroups

Research output

Subgroup induction property for branch groups

Weakly maximal subgroups of branch groups

Finitely generated subgroups of branch groups and subdirect products of just infinite groups

Activities

On the structure of finitely generated subgroups of branch groups

Maximal subgroups, weakly maximal subgroups and generalization

Sous-groupes maximaux, faiblement maximaux et généralisations

Cite this