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

Abstract

We use the relation between the volumes of the strata of mero- morphic quadratic differentials with at most simple poles on CP 1 and counting functions of the number of (bands of) closed geodesics in associated flat met- rics 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 Teichmuller geodesic flow we obtain quadratic asymptotics for the number of (bands of) closed trajectories and for the number of generalized diagonals in almost all right-angled billiards.