A Polytopal Decomposition of Strata of Translation Surfaces

Abstract

We study the decomposition of a stratum H(kappa) of translation surfaces into finitely many regions of surfaces that share a common L^infinity-Delaunay triangulation. In particular, we classify the infinitely many adjacencies between these isodelaunay regions. This classification allows us to construct a finite simplicial complex with the same homotopy type as H(kappa), and we outline a method for its computation. Finally, we show that Teichmueller curves also admit a decomposition into finitely many isodelaunay regions. Along the way, we require a stronger equivariant version of the traditional Nerve Lemma than currently exists in the literature, which we prove in the appendix.