Commensurated subgroups in finitely generated branch groups

TL;DR

This paper characterizes finitely generated branch groups as just infinite if and only if their commensurated subgroups are either finite or of finite index, with specific results for the Grigorchuk group.

Contribution

It establishes a new criterion linking the structure of commensurated subgroups to the just infinite property in finitely generated branch groups.

Findings

Finitely generated branch groups are just infinite iff all commensurated subgroups are finite or of finite index.

Every commensurated subgroup of the Grigorchuk group is either finite or of finite index.

Provides a characterization connecting subgroup properties to group infiniteness.

Abstract

A subgroup $\Delta\leq \Gamma$ is commensurated if $|\Delta:\Delta\cap \gamma\Delta\gamma^{-1}|<\infty$ for all $\gamma\in \Gamma$. We show a finitely generated branch group is just infinite if and only if every commensurated subgroup is either finite or of finite index. As a consequence, every commensurated subgroup of the Grigorchuk group is either finite or finite index.