On maximal subgroups of Thompson's group F

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.