The planar 3-colorable subgroup ε of Thompson’s group <i>F</i> and its even part

Abstract We study the planar 3-colorablesubgroup $\mathcal{E}$ 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.