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.