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

Abstract

A few years ago the so-called oriented subgroup [Formula: see text] of the Thompson group [Formula: see text] 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 [Formula: see text] it corresponds exactly to the oriented ones. Thanks to the work of Golan and Sapir, [Formula: see text] provided the first example of a maximal subgroup of infinite index in [Formula: see text] different from the parabolic subgroups that fix a point in [Formula: see text]. In this paper we investigate possible analogues of [Formula: see text] in higher Thompson groups [Formula: see text], with [Formula: see text], introduced by Brown. Most notably, we study algebraic properties of the oriented subgroup [Formula: see text] of [Formula: see text], as described recently by Jones, and prove in particular that it gives rise to a non-parabolic maximal subgroup of infinite index in [Formula: see text] and that the corresponding quasi-regular representation is irreducible.