Higher Rank Orbit Closures in $H^{odd}(4)$

The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component $H^{odd}(4)$ the only $GL^+(2,R)$ orbit closures are closed orbits, the Prym locus $\tilde{Q}(3,-1^3)$, and $H^{odd}(4)$. This is joint work with Duc-Manh Nguyen and Alex Wright.

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