Maximal subgroups of groups of intermediate growth

TL;DR

This paper investigates the maximal subgroups of infinite index in groups of intermediate growth acting on rooted binary trees, revealing they have countably many such subgroups, contrasting with other classes of groups.

Contribution

It provides the first classification of maximal subgroups of infinite index for a family of intermediate growth groups acting on trees, including non-torsion iterated monodromy groups.

Findings

Non-torsion iterated monodromy groups have countably many maximal subgroups of infinite index.

Groups of this family are just infinite and have the congruence subgroup property.

Contrasts with torsion groups and geometric groups regarding the number of such subgroups.

Abstract

Finding the number of maximal subgroups of infinite index of a finitely generated group is a natural problem that has been solved for several classes of `geometric' groups (linear groups, hyperbolic groups, mapping class groups, etc). Here we provide a solution for a family of groups with a different geometric origin: groups of intermediate growth that act on rooted binary trees. In particular, we show that the non-torsion iterated monodromy groups of the tent map (a special case of some groups first introduced by {\v{S}}uni{\'c} in \cite{Sunic} as `siblings of the Grigorchuk group') have exactly countably many maximal subgroups of infinite index, and describe them up to conjugacy. This is in contrast to the torsion case (e.g. Grigorchuk group) where there are no maximal subgroups of infinite index. It is also in contrast to the above-mentioned geometric groups, where there are either none or uncountably many such subgroups. Along the way we show that all the groups defined by {\v{S}}uni{\'c} have the congruence subgroup property and are just infinite.