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{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} .