2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62158We 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.48 pages, 14 figures. Revised final version. To appear, J. Amer. Math. SocGeometric TopologyDynamical Systems2000 MSC. Primary 30F40; Secondary 30F60, 37F30text