  We introduce a coarse combinatorial description of the Weil-Petersson
distance d_WP(X,Y) between two finite area hyperbolic Riemann surfaces X and Y.
The combinatorics reveal a connection between Riemann surfaces and hyperbolic
3-manifolds conjectured by Thurston: the volume of the convex core of the
quasi-Fuchsian manifold Q(X,Y) with X and Y in its boundary is comparable to
the Weil-Petersson distance d_WP(X,Y). Applications include a connection of the
Weil-Petersson distance with the Hausdorff dimension of the limit set and the
lowest eigenvalue of the Laplacian as well as a new finiteness criterion for
geometric limits.
