On maximal subgroups of infinite index in branch and weakly branch groups

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.

arXiv comment27 pages

References in corpus (4)

• Amenability via random walks
• Maximal subgroups of groups of intermediate growth
• Finitely presented groups associated with expanding maps
• Pro-$\mathcal{C}$ congruence properties for groups of rooted tree automorphisms

Cited by in corpus (5)

• Subgroup induction property for branch groups
• Maximal subgroups of non-torsion Grigorchuk-Gupta-Sidki groups
• $p$-Basilica groups
• On quasi-2-transitive actions of branch groups
• Maximal subgroups of a family of iterated monodromy groups