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

TL;DR

This paper investigates the structure and properties of the planar 3-colorable subgroup of Thompson's group F and its even part, revealing their stabilizer descriptions, closure properties, and representation theory.

Contribution

It provides a new description of the even part of the subgroup in terms of stabilizers and analyzes associated quasi-regular representations.

Findings

The even part is described via stabilizers of dyadic rationals.

The even part is shown to be closed in the sense of Golan and Sapir.

Two representations are irreducible, one is reducible.

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.