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The planar 3-colorable subgroup ε of Thompson’s group F and its even part
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society , First View
- Published online by Cambridge University Press:
- 25 November 2024, pp. 1-25
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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.
