Duke Mathematical Journal

$\operatorname{GL}_{2}\mathbb{R}$ orbit closures in hyperelliptic components of strata

Paul Apisa

Abstract

The object of this article is to study $\operatorname{GL}_{2}\mathbb{R}$ 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 $\operatorname{GL}_{2}\mathbb{R}$ 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

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

Dates
Revised: 2 June 2017
First available in Project Euclid: 30 January 2018

https://projecteuclid.org/euclid.dmj/1517281370

Digital Object Identifier
doi:10.1215/00127094-2017-0043

Mathematical Reviews number (MathSciNet)
MR3769676

Zentralblatt MATH identifier
06857028

Citation

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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