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The planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part

Valeriano Aiello, Tatiana Nagnibeda

TL;DR

The paper characterizes the planar $3$-colorable subgroup $\\mathcal{E}$ of Thompson's group $F$ and its even part $\\mathcal{E}_{\\rm EVEN}$, describing $\\mathcal{E}$ via generators $x_0$ and endomorphisms of the $\\mathcal{F}$ subgroup and identifying $\\mathcal{E}_{\\rm EVEN}$ as the intersection of stabilizers of three dyadic-subset partitions defined by a weight function. It proves $\\mathcal{E}_{\\rm EVEN}$ is closed in the sense of Golan–Sapir, determines its commensurator as $\\mathcal{E}$ within $F$, and establishes self-commensurability in the rectangular subgroups $K_{(2,1)}$ and $K_{(2,2)}$, yielding irreducibility results for the corresponding quasi-regular representations. The work connects these subgroups to Jones' representations arising from planar algebras, and provides concrete descriptions of when associated representations are irreducible or reducible. Overall, it advances the understanding of subgroup structure, stabilizer descriptions on dyadic rationals, and the representation theory of $F$ in the Jones framework, with explicit links to finite-index subgroups and closedness notions.

Abstract

We study the planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part $\mathcal{E}_{\rm EVEN}$. The latter is obtained by cutting $\mathcal{E}$ with a finite index subgroup of $F$ isomorphic to $F$, namely the rectangular subgroup $K_{(2,2)}$. We show that the even part $\mathcal{E}_{\rm EVEN}$ of the planar $3$-colorable subgroup admits a description in terms of stabilisers of suitable subsets of dyadic rationals. As a consequence $\mathcal{E}_{\rm EVEN}$ is closed in the sense of Golan and Sapir. We then study three quasi-regular representations associated with $\mathcal{E}_{\rm EVEN}$: two are shown to be irreducible and one to be reducible.

The planar $3$-colorable subgroup $\mathcal{E}$ of Thompson's group $F$ and its even part

TL;DR

The paper characterizes the planar -colorable subgroup of Thompson's group and its even part , describing via generators and endomorphisms of the subgroup and identifying as the intersection of stabilizers of three dyadic-subset partitions defined by a weight function. It proves is closed in the sense of Golan–Sapir, determines its commensurator as within , and establishes self-commensurability in the rectangular subgroups and , yielding irreducibility results for the corresponding quasi-regular representations. The work connects these subgroups to Jones' representations arising from planar algebras, and provides concrete descriptions of when associated representations are irreducible or reducible. Overall, it advances the understanding of subgroup structure, stabilizer descriptions on dyadic rationals, and the representation theory of in the Jones framework, with explicit links to finite-index subgroups and closedness notions.

Abstract

We study the planar -colorable subgroup of Thompson's group and its even part . The latter is obtained by cutting with a finite index subgroup of isomorphic to , namely the rectangular subgroup . We show that the even part of the planar -colorable subgroup admits a description in terms of stabilisers of suitable subsets of dyadic rationals. As a consequence is closed in the sense of Golan and Sapir. We then study three quasi-regular representations associated with : two are shown to be irreducible and one to be reducible.
Paper Structure (3 sections, 22 theorems, 45 equations, 6 figures)

This paper contains 3 sections, 22 theorems, 45 equations, 6 figures.

Table of Contents

  1. Preliminaries and notation
  2. The planar $3$-colorable subgroup and its even part
  3. Quasi-regular representations and closed subgroups

Key Result

Lemma 2.2

Let $f\in\mathcal{E}$. Then the following conditions are equivalent

Figures (6)

  • Figure 1: A tree diagram and its closure.
  • Figure 2: The generators of $F=F_2$.
  • Figure 3: The generators of the Brown-Thompson group $F_4$.
  • Figure 4: Ren's map $\Phi$ from the set of $4$-ary trees to binary trees.
  • Figure 5: The generators of $\mathcal{F}$.
  • ...and 1 more figures

Theorems & Definitions (46)

  • Example 2.1
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Example 2.5
  • Theorem 2.6
  • proof
  • Definition 2.7
  • ...and 36 more