On maximal subgroups in infinite finitely generated groups

Margulis and Soifer proved that a finitely generated linear group has all maximal subgroups of finite index (has the MF property) if and only if it is virtually solvable. Otherwise it has uncountably many maximal subgroups of infinite index. Pervova later proved that outside the world of linear groups Grigorchuk's group also has the MF property. However it is known to have a huge variety of weakly maximal subgroups. I will discuss some questions stemming from the aforementioned results about the richness versus rigidity of maximal and weakly maximal subgroups in various classes of groups, in particular in branch groups and in Thompson groups.