Generalizations of the Eierlegende-Wollmilchsau

TL;DR

This paper classifies a broad family of GL(2,R)-invariant subvarieties related to Teichmuller dynamics, providing foundational results that enable further classification of high-rank invariant subvarieties and applications to complex geometry.

Contribution

It offers a comprehensive classification of certain GL(2,R)-invariant subvarieties, including notable loci and orbits, advancing understanding in Teichmuller dynamics and complex geometry.

Findings

Classification of key invariant subvarieties including Eierlegende-Wollmilchsau and Ornithorynque

Construction of new examples that answer open questions negatively

Applications to the complex geometry of Teichmuller space

Abstract

We classify a natural collection of GL(2,R)-invariant subvarieties, which includes loci of double covers, the orbits of the Eierlegende-Wollmilchsau, Ornithorynque, and Matheus-Yoccoz surfaces, and loci appearing naturally in the study of the complex geometry of Teichmuller space. This classification is the key input in subsequent work of the authors that classifies "high rank" invariant subvarieties, and in subsequent work of the first author that classifies certain invariant subvarieties with "Lyapunov spectrum as degenerate as possible". We also derive applications to the complex geometry of Teichmuller space and construct new examples, which negatively resolve two questions of Mirzakhani and Wright and illustrate previously unobserved phenomena for the finite blocking problem.