Duke Mathematical Journal

GL2R orbit closures in hyperelliptic components of strata

Paul Apisa

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The object of this article is to study GL2R orbit closures in hyperelliptic components of strata of Abelian differentials. The main result is that all higher-rank affine invariant submanifolds in hyperelliptic components are branched covering constructions; that is, every translation surface in the affine invariant submanifold covers a translation surface in a lower genus hyperelliptic component of a stratum of Abelian differentials. This result implies the finiteness of algebraically primitive Teichmüller curves in all hyperelliptic components for genus greater than two. A classification of all GL2R orbit closures in hyperelliptic components of strata (up to computing connected components and up to finitely many nonarithmetic rank one orbit closures) is provided. Our main theorem resolves a pair of conjectures of Mirzakhani in the case of hyperelliptic components of moduli space.

Article information

Duke Math. J., Volume 167, Number 4 (2018), 679-742.

Received: 26 March 2016
Revised: 2 June 2017
First available in Project Euclid: 30 January 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37F30: Quasiconformal methods and Teichmüller theory; Fuchsian and Kleinian groups as dynamical systems
Secondary: 32G

translation surfaces Abelian differentials affine invariant submanifolds Teichmüller theory


Apisa, Paul. $\operatorname{GL}_{2}\mathbb{R}$ orbit closures in hyperelliptic components of strata. Duke Math. J. 167 (2018), no. 4, 679--742. doi:10.1215/00127094-2017-0043. https://projecteuclid.org/euclid.dmj/1517281370

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