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 ${\mathcal{E}}_{\mathrm{E}\mathrm{V}\mathrm{E}\mathrm{N}}$. 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 ${\mathcal{E}}_{\mathrm{E}\mathrm{V}\mathrm{E}\mathrm{N}}$ of the planar $3$-colorable subgroup admits a description in terms of stabilisers of suitable subsets of dyadic rationals. As a consequence ${\mathcal{E}}_{\mathrm{E}\mathrm{V}\mathrm{E}\mathrm{N}}$ is closed in the sense of Golan and Sapir. We then study three quasi-regular representations associated with ${\mathcal{E}}_{\mathrm{E}\mathrm{V}\mathrm{E}\mathrm{N}}$: two are shown to be irreducible and one to be reducible.