On Maximal Subgroups of Thompson's Group $F$

TL;DR

This paper investigates the structure of maximal subgroups in Thompson's group F, showing they are closed, undistorted, and constructing many examples of such subgroups, advancing understanding of subgroup dynamics in F.

Contribution

It proves that all infinite index maximal subgroups of F are closed and constructs numerous non-isomorphic examples, providing new insights into subgroup structure.

Findings

Maximal subgroups of infinite index in F are closed.

Finitely generated subgroups are contained in finitely generated maximal subgroups.

Constructed an infinite family of non-isomorphic maximal subgroups of infinite index.

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 $F$.