## Right-Angled Billiards and Volumes of the Moduli Spaces of Quadratic Differentials on a Riemann Sphere

#### Anton ZorichInstitut Mathématiques de Jussieu

We use the relation between the volumes of the strata of meromorphic quadratic differentials with at most simple poles on $CP^1$ and counting functions of the number of (bands of) closed geodesics in associated flat metrics with singularities to prove a very explicit formula for the volume of each such stratum conjectured by M. Kontsevich a decade ago. Applying ergodic techniques to the Teichm\"uller geodesic flow we obtain weak quadratic asymptotics for the number of (bands of) closed trajectories and for the number of generalized diagonals in almost all right-angled billiards. The results are obtianed in collaboration with J. Athreya and A. Eskin.

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