In the moduli space of Abelian differentials, big invariant subvarieties come from topology
INFORMAL GEOMETRY AND DYNAMICS
Paul Apisa - Yale University
It is a beautiful fact that any holomorphic one-form on a genus g Riemann surface can be presented as a collection of polygons in the plane with sides identified by translation. Since GL(2, R) acts on the plane (and polygons in it), it follows that there is an action of GL(2, R) on the collection of holomorphic one-forms on Riemann surfaces. This GL(2, R) action can also be described as the group action generated by scalar multiplication and Teichmuller geodesic flow. By work of McMullen in genus two, and Eskin, Mirzakhani, and Mohammadi in general, given any holomorphic one-form, the closure of its GL(2, R) orbit is an algebraic variety. While McMullen classified these orbit closures in genus two, little is known in higher genus.
In the first part of the talk, I will describe the Mirzakhani-Wright boundary of an invariant subvariety (using mostly pictures) and a new result about reconstructing an orbit closure from its boundary. In the second part of the talk, I will define the rank of an invariant subvariety - a measure of size related to dimension - and explain why invariant subvarieties of rank greater than g/2 are loci of branched covers of lower genus Riemann surfaces. This will address a question of Mirzakhani.
No background on Teichmuller theory or dynamics will be assumed. This material is work in progress with Alex Wright.