Simple geodesics on hyperbolic surfaces and the volume of the moduli space of curves [PhD Thesis]

The central result of this thesis is a recursive formula for the volume ,n(.el,. .., .en) of the moduli space Mg,n(.e 1 ,. .., .en) of hyperbolic Riemann surfaces of genus 9 with n geodesic boundary components. We show that ,n(.e) is a polynomial whose coefficients are rational multiples of powers of 1[. The constant term of the polynomial ,n(.e) is the Weil- Petersson volume of the traditional moduli space of closed surfaces of genus 9 with n marked points. We establish a relationship between the coefficients of volume polynomi- als and intersection numbers of tautological lines bundles over the moduli space of curves and show that the generating function for these intersection numbers satisfies the Virasoro equations. We also show that the number of simple closed geodesics of length < L on X E Mg,n has the asymptotic behavior sx(L) rv nxL6g-6+n as L 00. We relate the function nx to the geometry of X and intersection theory on moduli spaces of curves, and calculate the frequencies of different types of simple closed geodesics on a hyperbolic surface.