Right-angled billiards and volumes of moduli spaces of quadratic   differentials on $\mathbb{C}P^1$
  We use the relation between the volumes of the strata of meromorphic quadratic differentials with at most simple poles on the Riemann sphere 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.   The proof is based on a formula for the Lyapunov exponents of the geodesic flow on the moduli space, which gives a recursive formula from which the volumes could be recovered.   An appendix with an ergodic theorem proved by Jon Chaika is added to the current version. Applying this ergodic theorem to the Teichmueller geodesic flow we obtain EXACT quadratic asymptotics for the number of (bands of) closed trajectories and for the number of generalized diagonals in almost all right-angled billiards. All coefficients in the asymptotics (expressed in terms of the associated Siegel-Veech constants) are explicitly computed. 
Jayadev S. Athreya, Alex Eskin, and Anton Zorich
math.DS math.CO math.GT