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Counting closed loops in a stratum of quadratic differentials

4:00 pm Thursday, October 20, 2011
Kasra Rafi (University of Oklahoma)

In his thesis, Margulis used the mixing properties of the geodesic flow on a hyperbolic surface to find asymptotic growth rate for the number of closed geodesics of length less than R on a given surface. His argument has been emulated to many other settings. We examine the Teichmüller geodesic flow on the moduli space of surfaces, or more generally any stratum of quadratic differentials in cotangent bundle of moduli space. These flows are know to be mixing, but the spaces are not compact. However, we can show that the Teichmüller geodesic flow (or more precisely an associated random walk) 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.)

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