Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Valeriano Aiello, Tatiana Nagnibeda

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.

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

Paper Structure

This paper contains 5 sections, 38 theorems, 68 equations, 12 figures.

Table of Contents

  1. Preliminaries and notation
  2. The $3$-colorable subgroup as a stabiliser subgroup
  3. The rectangular subgroup $K_{(2,2)}$
  4. Maximal infinite index subgroups of $F$ containing an isomorphic image of $\mathcal{F}$
  5. On maximal infinite index subgroups of $F$

Key Result

Theorem 1.2

The map $\alpha_T$ with $T$ depicted in Figure fig-ren-map is an isomorphism between $F_4$ and the $3$-colorable subgroup. In particular, the $3$-colorable subgroup is generated by the following elements $w_0:=x_0^2x_1x_2^{-1}$, $w_1:=x_0x_1^2x_0^{-1}$, $w_2:=x_1^2x_3x_2^{-1}$, $w_3:=x_2^2x_3x_4^{-1

Figures (12)

  • Figure 1: Ren's map for $\vec{F}$.
  • Figure 2: The map for $\mathcal{F}$.
  • Figure 3: The generators of $F=F_2$.
  • Figure 4: The generators of $F_4$.
  • Figure 5: The generators of $\mathcal{F}$.
  • ...and 7 more figures

Theorems & Definitions (80)

  • Theorem 1.2: Ren
  • Lemma 1.3
  • proof
  • Lemma 1.4
  • proof
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 70 more