The Weil-Petersson metric and volumes of 3-dimensional hyperbolic convex cores

We introduce a coarse combinatorial description of the Weil-Petersson distance dWP(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 dWP(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. Figure 1. The lift to H of a quasi-Fuchsian convex core boundary component. ∗Research partially supported by NSF grant DMS-0072133 and an NSF postdoctoral fellowship.