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Abstract

The moduli space of holomorphic one-forms on Riemann surfaces admits a natural action by GL(2,R). This thesis is concerned with using the results of Eskin, Mirzakhani, and Mohammadi to study the orbit closures of points under this action. The first two chapters show that for hyperelliptic components of strata of Abelian differentials every orbit is closed, dense, or contained in a locus of branched covers. The final chapter studies orbits of translation surfaces with marked points and relates the results to rational billiards and the existence of holomorphic sections of the universal curve restricted to subvarieties of moduli space.

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