On maximal subgroups of Thompson's group F

We study subgroups of Thompson's group F by means of a 2-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 2-automaton associated with it. It follows that every finitely generated maximal subgroup of F is undistorted in F. We also demonstrate the construction of a maximal subgroup of F and prove that F has infinitely many non-isomorphic maximal subgroups.