Teichmüller curves via the Hurwitz-Hecke construction

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

Given a Teichmüller curve, there is a way to construct another, in higher genus, by taking a branched cover. If a Teichmüller curve does not arise in this way, 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 two there exists a complete classification. In every genus starting from five 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 and Luke Jeffreys.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

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Geometry and topology online

Series comments: You can also find up-to-date information on the seminar homepage - warwick.ac.uk/fac/sci/maths/research/events/seminars/areas/geomtop/

The talks start at 13:30. Talks are typically fifty minutes long, with ten minutes for questions.

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