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

Abstract

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 $F_k,k\ge 2$, 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.

arXiv commentadded Acknowledgements

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Cited by in corpus (5)

• On the $3$-colorable subgroup $\mathcal{F}$ and maximal subgroups of Thompson's group $F$
• An introduction to Thompson knot theory and to Jones subgroups
• An $n$-adic generalization of the Lodha-Moore group
• An extension of Krishnan's central limit theorem to the Brown-Thompson groups
• The planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part