On the oriented Thompson subgroup F → 3 and its relatives in higher Brown-Thompson groups

A few years ago the so-called oriented subgroup F→ of the Thompson group F was introduced by V. Jones while investigating the connections between subfactors and conformal field theories. In the coding of links and knots by elements of F it corresponds exactly to the oriented ones. Thanks to the work of Golan and Sapir, F→ provided the first example of a maximal subgroup of infinite index in F different from the parabolic subgroups that fix a point in (0, 1). In this paper we investigate possible analogues of F→ in higher Thompson groups Fk,k ≥ 2, with F = F2, introduced by Brown. Most notably, we study algebraic properties of the oriented subgroup F→3 of F3, as described recently by Jones, and prove in particular that it gives rise to a non-parabolic maximal subgroup of infinite index in F3 and that the corresponding quasi-regular representation is irreducible.