A few years ago the so-called oriented subgroup $\overrightarrow{F}$ 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, $\overrightarrow{F}$ 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 $\overrightarrow{F}$ in higher Thompson groups $F_k,k\ge 2$, with $F=F_2$, introduced by Brown. Most notably, we study algebraic properties of the oriented subgroup ${\overrightarrow{F}}_3$ 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.

几年前所谓的定向子群$\overrightarrow{F}$汤普森集团$F$由 V. Jones 在研究子因子和共形场理论之间的联系时介绍。在通过元素对链接和结进行编码时$F$它与定向的完全对应。感谢 Golan 和 Sapir 的工作，$\overrightarrow{F}$提供了无限索引的最大子群的第一个例子$F$不同于固定点的抛物线子群$(0,1)$. 在本文中，我们研究了可能的类似物$\overrightarrow{F}$在更高汤普森组$F_{ķ},ķ\ge 2$， 和$F=F_2$，由布朗介绍。最值得注意的是，我们研究了有向子群的代数性质${\overrightarrow{F}}_3$的$F_3$，正如琼斯最近描述的那样，并特别证明它产生了一个无限指数的非抛物线最大子群$F_3$并且相应的准正则表示是不可约的。