## 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