15 March 2018 GL2R orbit closures in hyperelliptic components of strata
Paul Apisa
Duke Math. J. 167(4): 679-742 (15 March 2018). DOI: 10.1215/00127094-2017-0043


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.


Download Citation

Paul Apisa. "GL2R orbit closures in hyperelliptic components of strata." Duke Math. J. 167 (4) 679 - 742, 15 March 2018. https://doi.org/10.1215/00127094-2017-0043


Received: 26 March 2016; Revised: 2 June 2017; Published: 15 March 2018
First available in Project Euclid: 30 January 2018

zbMATH: 06857028
MathSciNet: MR3769676
Digital Object Identifier: 10.1215/00127094-2017-0043

Primary: 37F30
Secondary: 32G

Keywords: Abelian differentials , affine invariant submanifolds , Teichmüller theory , translation surfaces

Rights: Copyright © 2018 Duke University Press


This article is only available to subscribers.
It is not available for individual sale.

Vol.167 • No. 4 • 15 March 2018
Back to Top