We show that every orbit closure of translation surfaces is a connected component of a stratum, the hyperelliptic locus, or consists entirely of surfaces whose Jacobians have extra endomorphisms. We use this result to give applications related to polygonal billiards. For example, we exhibit infinitely many rational triangles whose unfoldings have dense orbit.
"Full-rank affine invariant submanifolds." Duke Math. J. 167 (1) 1 - 40, 15 January 2018. https://doi.org/10.1215/00127094-2017-0036