Since October 1, I am at the Goethe-Universität Frankfurt/Main.
|Summer Semester 2011||Algebraic Geometry II||Lecture|
Origamis are translation surfaces obtained by gluing finitely many unit squares. These combinatorial objects provide an easy access to Teichmüller curves - algebraic curves in the moduli space of curves. In particular, their monodromy represenation, the action of the fundamental group of the Teichmüller curve on the cohomology of the fibre, can be explicitely determined. In my Ph.D. thesis, a general principle for the decomposition of this represenation is exhibited and then applied to examples. Closely connected to it is a dynamical cocycle, the Kontsevich-Zorich cocycle on the Teichmüller curve. By the work of M. Kontsevich, the Lyapunov exponents that govern the dynamics of the cocycle are related to degrees of certain line bundles on the Teichmüller curve. Using this relationship, it is shown that the Lyapunov exponents, otherwise inaccessible, can be computed in the case of a subrepresentation of rank two.
You will find more on origamis on the webpage origamis in Karlsruhe.
- A. Kappes, On the Equation of an Origami of Genus two with two Cusps, Diploma thesis, (last changes: April 10, 2007) (.pdf-file)
- F. Herrlich, A. Kappes, G. Schmithüsen, An origami of genus 2 with a translation, preprint (2008) (.pdf-file)
- A. Kappes, Monodromy Representations and Lyapunov Exponents of Origamis, Ph.D. thesis, May 2011 (.pdf-Datei)