On the $3$-colorable subgroup $\mathcal{F}$ and maximal subgroups of Thompson's group $F$

Abstract

In his work on representations of Thompson's group $F$, Vaughan Jones defined and studied the $3$-\emph{colorable subgroup} $\mathcal{F}$ of $F$. Later, Ren showed that it is isomorphic with the Brown-Thompson group $F_4$. In this paper we continue with the study of the $3$-colorable subgroup and prove that the quasi-regular representation of $F$ associated with the $3$-colorable subgroup is irreducible. We show moreover that the preimage of $\mathcal{F}$ under a certain injective endomorphism of $F$ is contained in three (explicit) maximal subgroups of $F$ of infinite index. These subgroups are different from the previously known infinite index maximal subgroups of $F$, namely the parabolic subgroups that fix a point in $(0,1)$, (up to isomorphism) the Jones' oriented subgroup $\vec{F}$, and the explicit examples found by Golan.

arXiv commentWe removed Theorem 2.9, which was incorrect, and changed the proof of Theorem 2.10 accordingly

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Cited by in corpus (4)

• The $3$-colorable subgroup of Thompson's group and tricolorability of links
• An introduction to Thompson knot theory and to Jones subgroups
• 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