On the oriented Thompson subgroup (F)over-right-arrow(3) and its relatives in higher Brown-Thompson groups

A few years ago the so-called oriented subgroup (F) over right arrow 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) over right arrow 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) over right arrow in higher Thompson groups F-k, k >= 2, with F = F-2, introduced by Brown. Most notably, we study algebraic properties of the oriented subgroup (F) over right arrow (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.