Right-angled billiards and volumes of moduli spaces of quadratic differentials on $\mathbb{C}P^1$

@article{Athreya2012RightangledBA,
title={Right-angled billiards and volumes of moduli spaces of quadratic differentials on \$\mathbb\{C\}P^1\$},
author={J. Athreya and A. Es'kin and A. Zorich},
journal={arXiv: Dynamical Systems},
year={2012}
}

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… CONTINUE READING