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On Maximal Subgroups of Thompson's Group $F$
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abstract
We study subgroups of Thompson's group $F$ by means of an automaton associated with them. We prove that every maximal subgroup of $F$ of infinite index is closed, that is, it coincides with the subgroup of $F$ accepted by the automaton associated with it. It follows that every finitely generated maximal subgroup of $F$ is undistorted in $F$. We also prove that every finitely generated subgroup of $F$ is contained in a finitely generated maximal subgroup of $F$ and construct an infinite family of non-isomorphic maximal subgroups of infinite index in $F$.
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Higman--Thompson groups $F_n$ all the way down
Every Higman–Thompson group Fn admits a chain of maximal infinite-index copies of itself with trivial intersection, realized by semi-synchronizing transducers.
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