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

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 while the natural action of $\mathcal{E}$ on the dyadic rationals is transitive, its even part $\mathcal{E}_{\rm EVEN}$ admits a description in terms of stabilisers of suitable subsets of dyadic rationals. As a consequence both $\mathcal{E}_{\rm EVEN}$ and $\mathcal{E}$ are 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.