The $L^\infty$-isodelaunay decomposition of strata of abelian differentials

Abstract

We study the decomposition of a stratum $\mathcal H(\kappa)$ of abelian differentials into regions of differentials that share a common $L^\infty$-Delaunay triangulation. In particular, we classify the infinitely many adjacencies between these isodelaunay regions, a phenomenon whose observation is attributed to Filip in work of Frankel. This classification allows us to construct a finite simplicial complex with the same homotopy type as $\mathcal H(\kappa)$, and we outline a method for its computation. We also require a stronger equivariant version of the traditional Nerve Lemma than currently exists in the literature, which we prove.