Classification of higher rank orbit closures in ${\mathcal H^{\mathrm{odd}}(4)}$

Abstract

The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component ${\mathcal H^{\mathrm{odd}}(4)}$ the only ${GL^+(2,\mathbb R)}$ orbit closures are closed orbits, the Prym locus ${\tilde{\mathcal{Q}}(3,-1^3)}$, and ${\mathcal H^{\mathrm{odd}}(4)}$. Together with work of Matheus–Wright, this implies that there are only finitely many non-arithmetic closed orbits (Teichmüller curves) in $\mathcal H^{\mathrm{odd}}(4)$ outside of the Prym locus.