## Duke Mathematical Journal

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

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

#### Abstract

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

Received: 26 March 2016

Revised: 2 June 2017

First available in Project Euclid: 30 January 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/00127094-2017-0043

**Mathematical Reviews number (MathSciNet)**

MR3769676

**Zentralblatt MATH identifier**

06857028

**Subjects**

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

Secondary: 32G

**Keywords**

translation surfaces Abelian differentials affine invariant submanifolds Teichmüller theory

#### 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