[Paper Review] On maximal subgroups of infinite index in branch and weakly branch groups

Dominik Francoeur|arXiv (Cornell University)|Apr 3, 2020

Finite Group Theory Research19 references13 citations

TL;DR

This paper generalizes a technical tool from Pervova's work on maximal subgroups in Grigorchuk and GGS groups to all weakly branch groups satisfying a natural condition, proving that every maximal subgroup of infinite index in a branch group is itself a branch group. As a key application, it shows that the Basilica group—though not a branch group—has no maximal subgroups of infinite index, meaning all its maximal subgroups are of finite index, making it the first known weakly branch group with this property.

ABSTRACT

We generalise a technical tool, originally developed by Pervova for the study of maximal subgroups in Grigorchuk and GGS groups, to all weakly branch groups satisfying a natural condition, and in particular to all branch groups. We then use this tool to prove that every maximal subgroup of infinite index of a branch group is also a branch group. As a further application of this result, we show that every maximal subgroup of the Basilica group is of finite index.

Motivation & Objective

• To extend Pervova's technical tool on projections of vertex stabilizers to all weakly branch groups satisfying a natural condition.
• To prove that every maximal subgroup of infinite index in a branch group is itself a branch group.
• To analyze the maximal subgroups of the Basilica group, a weakly branch but not branch group, to determine whether they can have infinite index.
• To establish that the Basilica group has no maximal subgroups of infinite index, implying all its maximal subgroups are of finite index.
• To demonstrate that the Basilica group is the first known example of a weakly branch group (not a branch group) with no maximal subgroups of infinite index.

Proposed method

• Generalize a technical lemma from Pervova's work on Grigorchuk and GGS groups to all weakly branch groups satisfying a natural condition, particularly those with regular and primitive action on the first level of a rooted tree.
• Use the generalized tool to show that projections of maximal subgroups of infinite index in branch groups are also maximal subgroups of infinite index in the corresponding vertex groups.
• Prove that if a maximal subgroup of infinite index exists in a branch group, then its vertex stabilizers must also be maximal subgroups of infinite index, leading to a contradiction unless the subgroup is itself a branch group.
• Apply the generalized tool to the Basilica group by analyzing prodense subgroups and their projections to vertex groups.
• Use the fact that every proper quotient of the Basilica group lies in the class of groups with finite maximal subgroups (MF) to show that any maximal subgroup of infinite index would contradict the projection result.
• Combine the projection result with the structure of the Basilica group to conclude that no maximal subgroup of infinite index can exist.

Experimental results

Research questions

• RQ1 Can Pervova’s method for studying maximal subgroups in Grigorchuk and GGS groups be generalized to all weakly branch groups satisfying a natural condition?
• RQ2 Is every maximal subgroup of infinite index in a branch group necessarily a branch group itself?
• RQ3 Does the Basilica group, a weakly branch but not branch group, admit any maximal subgroups of infinite index?
• RQ4 What structural properties do prodense subgroups of the Basilica group have, particularly regarding their projections to vertex groups?
• RQ5 Can the absence of maximal subgroups of infinite index in the Basilica group be established using the generalized projection technique?

Key findings

• The paper establishes a generalization of Pervova’s technical tool to all weakly branch groups satisfying a natural condition, including all branch groups.
• It proves that every maximal subgroup of infinite index in a branch group is itself a branch group, providing a structural characterization of such subgroups.
• The Basilica group has no maximal subgroups of infinite index, meaning all its maximal subgroups are of finite index.
• This makes the Basilica group the first known example of a weakly branch but not branch group with no maximal subgroups of infinite index.
• The result shows that the class of groups with no maximal subgroups of infinite index extends beyond just-infinite branch groups to include certain weakly branch groups with different algebraic properties, such as being torsion-free and not just-infinite.