Abstract: A closed surface can be endowed with a certain locally Euclidean metric structure called a translation surface. Moduli spaces that parametrize such structures are called strata, and there is still much to discover of their global topology. These strata admit a decomposition into finitely many polytopal regions parametrized by certain triangulations of translation surfaces (L-infinity Delaunay triangulations). These regions intersect each other in pathological ways (the "infinite adjacency phenomenon"), but we resolve these pathologies to obtain finite simplicial models for strata. Our methods also show that there is an induced polytopal decomposition on subvarieties of strata called Teichmüller curves.

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