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

Erwan Lanneau and Duc-Manh Nguyen

Geometry & Topology 20 (2016) 1359–1426

This paper is devoted to the classification of GL+(2, )–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+(2, )–orbits (not necessarily primitive) in the Prym eigenform locus ΩED(2,2) for any fixed D that is not a square.

abelian differential, moduli spaces, orbit closure, real multiplication, Prym locus, translation surface
Mathematical Subject Classification 2010
Primary: 30F30, 32G15, 37D40, 54H20, 57R30
Received: 20 March 2014
Revised: 4 June 2015
Accepted: 20 July 2015
Published: 4 July 2016
Proposed: Jean-Pierre Otal
Seconded: Benson Farb, Martin Robert Bridson
Erwan Lanneau
Institut Fourier
Université Grenoble Alpes
BP 74
38402 Saint-Martin d’Hères
Duc-Manh Nguyen
Institut de Mathematiques de Bordeaux
Universite Bordeaux 1
Bat. A33
351 Cours de la Libération
33405 Bordeaux Talence Cedex