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 H^{odd}(4) the only GL^+(2,R) orbit closures are closed orbits, the Prym locus Q(3,-1^3), and H^{odd}(4). Together with work of Matheus-Wright, this implies that there are only finitely many non-arithmetic closed orbits (Teichmuller curves) in H^{odd}(4) outside of the Prym locus.
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