Maximal subgroups of a family of iterated monodromy groups (Q6648697)

Branch groups are groups acting level-transitively on a spherically homogeneous rooted tree \(\mathcal{T}\) and having subnormal subgroups similar to that of the full automorphism group \(\mathrm{Aut}(\mathcal{T})\) of the tree \(\mathcal{T}\). In this paper, the authors study a more general class of groups, the weakly branch groups, obtained by weakening some of the algebraic properties of branch groups.\N\NThe Basilica group is a \(2\)-generated weakly branch, but not branch, group acting on the binary rooted tree. A more general form of the Basilica group has been investigated by \textit{J. M. Petschick} and \textit{K. Rajeev} ([Groups Geom. Dyn. 17, No. 1, 331--384 (2023; Zbl 1511.20093)]), which is an \(s\)-generated weakly branch, but not branch, group that acts on the \(m\)-adic tree, for \(s, m \geq 2\). A larger family of groups, which contains these generalised Basilica groups, is the family of iterated monodromy groups. An interesting problem is the study of the existence of maximal subgroups of infinite index in weakly branch groups ([\textit{D. Francoeur} and \textit{A. Garrido}, Adv. Math. 340, 1067--1107 (2018; Zbl 1499.20072)]). The reviewer points out that in the torsion case (e.g. Grigorchuk group) there are no maximal subgroups of infinite index.\N\NThe main result in the paper under review is that a subfamily of iterated monodromy groups, which more closely resemble the generalised Basilica groups, have maximal subgroups only of finite index.