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

TL;DR

This paper investigates the $3$-colorable subgroup of Thompson's group $F$, proving its associated representation is irreducible and identifying new maximal subgroups containing its preimage, expanding understanding of $F$'s subgroup structure.

Contribution

It demonstrates the irreducibility of the quasi-regular representation of $F$ related to the $3$-colorable subgroup and identifies three new maximal subgroups of $F$ of infinite index.

Findings

The quasi-regular representation of $F$ for the $3$-colorable subgroup is irreducible.

The preimage of $$ under an injective endomorphism lies in three new maximal subgroups of $F$.

These maximal subgroups are distinct from previously known infinite index maximal subgroups.

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.