Maximal subgroups in torsion branch groups

Abstract

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