Right-angled billiards and volumes of moduli spaces of quadratic differentials on $CP^1$

Lyapunov exponents of the Hodge bundle over the Teichm\"uller flow are expressed in terms of the Siegel-Veech constant, which is responsible for the growth rate of the number of closed geodesics on almost any flat surface in the associate family. The Siegel-Veech constant can be expressed in terms of volumes of the moduli spaces of flat surfaces.

I will show how these relations allow to compute volumes of the moduli spaces of meromorphic quadratic differentials with at most simple poles in genus zero (or, what is the same, compute the asymptotics of the number of pillowcase covers of genus zero with a fixed ramification profile). As an application, we get the counting formulae for the number of closed trajectories of right-angled billiards, and compute the diffusion rate for some wind-tree billiards. This is a joint work with Jayadev Athreya and Alex Eskin; applications to the windtree - work in progress with V. Delecroix.

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