Maximal subgroups of a family of iterated monodromy groups

TL;DR

This paper investigates the subgroup structure of a family of iterated monodromy groups, showing that certain subfamilies have only finite index maximal subgroups, extending previous results from branch to weakly branch groups.

Contribution

It demonstrates that a specific subfamily of iterated monodromy groups, resembling generalized Basilica groups, possess only finite index maximal subgroups, advancing understanding of subgroup properties in weakly branch groups.

Findings

Certain iterated monodromy groups have only finite index maximal subgroups.

Extension of subgroup analysis from branch to weakly branch groups.

Broader understanding of subgroup structures in complex group families.

Abstract

The Basilica group is a well-known 2-generated weakly branch, but not branch, group acting on the binary rooted tree. Recently a more general form of the Basilica group has been investigated by Petschick and Rajeev, which is an $s$-generated weakly branch, but not branch, group that acts on the $m$-adic tree, for $s,m > 1$. A larger family of groups, which contains these generalised Basilica groups, is the family of iterated monodromy groups. With the new developments by Francoeur, the study of the existence of maximal subgroups of infinite index has been extended from branch groups to weakly branch groups. Here we show that a subfamily of iterated monodromy groups, which more closely resemble the generalised Basilica groups, have maximal subgroups only of finite index.