04/26: Paul Apisa
In the moduli space of translation surfaces, large orbit closures are strata or loci of double covers
Abstract: Any translation surface can be presented as a collection of polygons in the plane with sides identified. By acting linearly on the polygons, we obtain an action of GL(2,R) on moduli spaces of translation surfaces. Recent work of Eskin, Mirzakhani, and Mohammadi showed that GL(2,R) orbit closures are locally described by linear equations on the edges of the polygons. However, which linear manifolds arise this way is mysterious.
In this talk, I will describe new joint work with Alex Wright that shows that when an orbit closure is sufficiently large it must be a whole moduli space, called a stratum in this context, or a locus defined by rotation by pi symmetry. We define “sufficiently large” in terms of rank, which is the most important numerical invariant of an orbit closure, and is an integer between 1 and the genus g. Our result applies when the rank is at least 1+g/2, and so handles roughly half of the possible values of rank.
I will conclude by explaining how the ideas that appear in the proof of the previous result have connections to studying the geometry of Teichmuller space and the Lyapunov spectrum of the Kontsevich-Zorich cocycle – the cocycle whose exponents govern how cohomology classes grow along Teichmuller geodesic flow.