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.