Commensurated subgroups in finitely generated branch groups

Abstract

A subgroup $\Delta \le \Gamma$ 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.

arXiv commentAccepted version