Geometry, Topology and Dynamics Seminar
Matthew Bainbridge
University of Chicago
Algebraically primitive Teichmüller curves in genus three
Abstract: A Teichmüller curve is a finite-volume Riemann surface $C = \mathbb{H}/G$ together with an isometric immersion $C \to M_g$, where $M_g$ is the moduli space of genus-$g$ Riemann surfaces, equipped with the Teichmüller metric. The trace field $F(C)$ is the number field obtained by adjoining to $\mathbb{Q}$ the traces of the elements of $G$ (a subgroup of $SL(2,\mathbb{R}))$. It is known that $F(C)$ has degree at most $g$, and $C$ is said to be algebraically primitive if $F(C)$ has degree $g$.
McMullen and Calta have independently discovered an infinite family of algebraically primitive Teichmüller curves in $M_2$. It is an open question whether there are infinitely many algebraically primitive Teichmüller curves in $M_g$ for any $g>2$. In this talk I will present some recent theorems and computer evidence indicating there should be only finitely many algebraically primitive Teichmüller curves in $M_3$. This is joint work with Martin Möller.
Wednesday April 15, 2009 at 3:00 PM in SEO 612