Philipp Bader

Teichmüller curves via the Hurwitz-Hecke construction

Teichmüller curves are totally geodesic algebraic curves inside the moduli space of Riemann surfaces. There are fascinating connections between Teichmüller curves and billiard flows on polygons.

Given a Teichmüller curve, one can always construct another one in higher genus by taking a branched cover. If a Teichmüller curve does not arise as a branched cover of a smaller genus Teichmüller curve, we call it primitive. The classification of primitive Teichmüller curves is a problem that has been widely explored in the past decades but still leaves many questions unanswered. In fact, only in genus 2 there exists a complete classification. In every genus starting from 5 and higher only finitely many examples of primitive Teichmüller curves have been found.

In this talk, we introduce the notions described above and present the so-called Hurwitz-Hecke construction; a method that can be used to construct Teichmüller curves. We will see that this construction gives rise to many of the known examples of Teichmüller curves. This is joint work in progress with Paul Apisa.