Connected components of the strata of the moduli space of meromorphic differentials

Abstract

We study the translation surfaces corresponding to meromorphic di fferentials on compact Riemann surfaces. Such geometric structures naturally appear when studying compactifications of the strata of the moduli space of Abelian di fferentials. We compute the number of connected components of the strata of the moduli space of meromorphic di fferentials. We show that in genus greater than or equal to two, one has up to three components with a similar description as the one of Kontsevich–Zorich for the moduli space of Abelian di fferentials. In genus one, one can obtain an arbitrarily large number of connected components that are distinguished by a simple topological invariant.