Cambridge Journal of Mathematics

Volume 10 (2022)

Number 4

Generalizations of the Eierlegende–Wollmilchsau

Pages: 859 – 933



Paul Apisa (Department of Mathematics, University of Wisconsin, Madison, Wisc., U.S.A.)

Alex Wright (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)


We classify a natural collection of $GL(2,\mathbb{R})$-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 Teichmüller 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 Teichmüller space and construct new examples, which negatively resolve two questions of Mirzakhani and Wright and illustrate previously unobserved phenomena for the finite blocking problem.

The first author was partially supported by NSF Postdoctoral Fellowship DMS 1803625, and the second author was partially supported by a Clay Research Fellowship, NSF Grant DMS 1856155, and a Sloan Research Fellowship.

Received 9 April 2021

Published 21 October 2022