  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.
