GL(2,R) Orbit Closures in Hyperelliptic Components of Strata

TL;DR

This paper classifies GL(2,R) orbit closures in hyperelliptic components of strata of abelian differentials, showing they are branched covering constructions and resolving key conjectures in the field.

Contribution

It proves all higher rank affine invariant submanifolds in hyperelliptic components are branched coverings, advancing the understanding of orbit closures and Teichmüller curves.

Findings

All higher rank affine invariant submanifolds are branched coverings.

Finiteness of algebraically primitive Teichmüller curves in hyperelliptic components for genus > 2.

Complete classification of GL(2,R) orbit closures in hyperelliptic components.

Abstract

The object of this paper is to study GL(2,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, i.e. 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 finiteness of algebraically primitive Teichmuller curves in all hyperelliptic components for genus greater than two. A classification of all GL(2, 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.