ON THE 3-COLORABLE SUBGROUP F and MAXIMAL SUBGROUPS OF THOMPSON’S GROUP F

In his work on representations of Thompson’s group F, Vaughan Jones defined and studied the 3-colorable subgroup F of F. Later, Ren showed that it is isomorphic to the Brown–Thompson group F4. In this paper we continue with the study of the 3-colorable subgroup and prove that the quasi-regular representation of F associated with the 3-colorable subgroup is irreducible. We show in addition that the preimage of F under a certain injective endomorphism of F is contained in three (explicit) maximal subgroups of F of infinite index. These subgroups are different from the previously known infinite index maximal subgroups of F, namely the parabolic subgroups that fix a point in (0, 1), (up to isomorphism) the Jones’ oriented subgroup F⃗ , and the explicit examples found by Golan.