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$\mathrm{GL}^+(2,\mathbb{R})$–orbits
in Prym eigenform loci
Erwan Lanneau and Duc-Manh Nguyen |
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Geometry & Topology 20 (2016) 1359–1426
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Abstract |
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This paper is devoted to the classification of –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 –orbits (not necessarily primitive) in the Prym eigenform locus for any fixed that is not a square. |
Keywords
abelian differential, moduli spaces, orbit closure, real
multiplication, Prym locus, translation surface
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Mathematical Subject Classification 2010
Primary: 30F30, 32G15, 37D40, 54H20, 57R30
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References |
Publication
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
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Authors |
| Erwan Lanneau | |
| Institut Fourier Université Grenoble Alpes BP 74 38402 Saint-Martin d’Hères France |
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| Duc-Manh Nguyen | |
| Institut de Mathematiques de
Bordeaux Universite Bordeaux 1 Bat. A33 351 Cours de la Libération 33405 Bordeaux Talence Cedex France |
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