Special Colloquium
Kasra Rafi
University of Oklahoma
Counting closed loops in a stratum of quadratic differentials
Abstract: In his thesis, Margulis computed the asymptotic growth rate
for the number of closed geodesics of length less than R on a given
closed hyperbolic surface and his argument has been emulated to many
other settings. We examine the Teichmüller geodesic flow on the moduli
space of a surface, or more generally any stratum of quadratic differentials
in the cotangent bundle of moduli space. The flow is known to be mixing,
but the spaces are not compact and the flow is not uniformly hyperbolic.
We show that the random walk associated to the Teichmüller geodesic
flow is biased toward the compact part of the stratum. We then use this
to find asymptotic growth rate of for the number of closed loops in the
stratum. (This is a joint work with Alex Eskin and Maryam Mirzakhani.)
Tuesday January 17, 2012 at 3:00 PM in SEO 636