Commensurated subgroups in finitely generated branch groups

A subgroup $Δ\leq Γ$ is commensurated if $|Δ:Δ\cap γΔγ^{-1}|<\infty$ for all $γ\in Γ$. 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.

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