Weakly maximal subgroups in regular branch groups

TL;DR

The paper demonstrates that regular branch groups have uncountably many weakly maximal subgroups containing any finite subgroup, including many that are not parabolic, expanding understanding of subgroup structures in these groups.

Contribution

It shows the existence of uncountably many automorphism classes of weakly maximal subgroups in regular branch groups, including non-parabolic ones, for the first time.

Findings

Uncountably many weakly maximal subgroups contain any finite subgroup Q.

Existence of uncountably many non-parabolic weakly maximal subgroups.

Application to Grigorchuk-Gupta-Sidki type groups.

Abstract

Let $G$ be a finitely generated regular branch group acting by automorphisms on a regular rooted tree $T$. It is well-known that stabilizers of infinite rays in $T$ (aka parabolic subgroups) are weakly maximal subgroups in $G$, that is, maximal among subgroups of infinite index. We show that, given a finite subgroup $Q\leq G$, $G$ possesses uncountably many automorphism equivalence classes of weakly maximal subgroups containing $Q$. In particular, for Grigorchuk-Gupta-Sidki type groups this implies that they have uncountably many automorphism equivalence classes of weakly maximal subgroups that are not parabolic.