Configurations of saddle connections of quadratic differentials on ${\mathbb{CP}}^1$ and on hyperelliptic Riemann surfaces

Abstract

Configurations of rigid collections of saddle connections are connected component invariants for strata of the moduli space of quadratic differentials. They have been classified for strata of Abelian differentials by Eskin, Masur and Zorich. Similar work for strata of quadratic differentials has been done by Masur and Zorich, although in that case the connected components were not distinguished.

We classify the configurations for quadratic differentials on $C{P}^1$ and on hyperelliptic connected components of the moduli space of quadratic differentials. We show that, in genera greater than five, any configuration that appears in the hyperelliptic connected component of a stratum also appears in the non-hyperelliptic one. For such genera, this enables to classify the configurations that appear for each connected component of each stratum.