$\mathrm{GL}^+(2,\mathbb{R})$–orbits in Prym eigenform loci

Abstract

This paper is devoted to the classification of ${GL}^{+}\left(2,ℝ\right)$–orbit closures of surfaces in the intersection of the Prym eigenform locus with various strata of abelian differentials. We show that the following dichotomy holds: an orbit is either closed or dense in a connected component of the Prym eigenform locus.

The proof uses several topological properties of Prym eigenforms. In particular, the tools and the proof are independent of the recent results of Eskin and Mirzakhani and Eskin, Mirzakhani and Mohammadi.

As an application we obtain a finiteness result for the number of closed ${GL}^{+}\left(2,ℝ\right)$–orbits (not necessarily primitive) in the Prym eigenform locus $\Omega E_D\left(2,2\right)$ for any fixed $D$ that is not a square.