On the oriented Thompson subgroup $\vec{F}_3$ and its relatives in higher Brown-Thompson groups

A few years ago the so-called oriented subgroup $\vec F$ 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, $\vec F$ 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 $\vec F$ in higher Thompson groups $F_k, k\geq 2$, with $F=F_2$, introduced by Brown. Most notably, we study algebraic properties of the oriented subgroup $\vec{F}_3$ of $F_3$, as described recently by Jones, and prove in particular that it gives rise to a non-parabolic maximal subgroup of infinite index in $F_3$ and that the corresponding quasi-regular representation is irreducible.