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.