Maximal subgroups in torsion branch groups

TL;DR

This paper investigates the structure of maximal subgroups in branch groups, providing criteria for inclusion in a class with finite index maximal subgroups, constructing groups with novel properties, and presenting new examples outside this class.

Contribution

It introduces criteria for classifying branch groups within  and constructs new examples, including a periodic branch group outside , with explicit maximal subgroup descriptions.

Findings

Identified criteria for groups in  with finite index maximal subgroups.

Constructed branch groups with non-normal maximal subgroups.

Presented explicit examples of branch groups outside , including a periodic one.

Abstract

We study the maximal subgroups of branch groups and obtain a criterion that ensures that certain spinal groups are contained in the class $\mathcal{MF}$ of groups with all maximal subgroups of finite index. This allows us to construct branch groups within $\mathcal{MF}$ exhibiting novel properties, for example groups that possess non-normal maximal subgroups. Furthermore, we give new concrete examples of branch groups outside $\mathcal{MF}$, with explicitly given maximal subgroups of infinite index. Most prominently, we construct a periodic branch group outside $\mathcal{MF}$.