Texas A&M University, Department of
Mathematics, 216 Milner Hall, 30th of April 2008, 4:00-4:50
Groups and Dynamics Seminar
Quadratic growth rates via modular fibers
Martin Schmoll of Clemson University
Recent structural results made it possible to evaluate quadratic growth
rates of certain types of geodesic segments on translation surfaces
using the Siegel-Veech formula. Roughly speaking, the Siegel-Veech
formula relates quadratic growth rates of a particular translation
surface to volume of its SL(2,R)-orbit closure (in moduli space) and
volumes of some ``lower dimensional" SL(2,R)-orbits in that closure.
This implies we have to understand SL(2,R)-orbit closures in moduli
space. Even for branched torus covers that is not easy. For instance,
results of McMullen classify all SL(2,R)-orbit closures in genus 2,
away from certain torus covers.
We introduce some ideas to gain information on torus covers and
branched covers in general and report on results obtained using these
ideas.