Let G be a geometrically finite Kleinian group. Its convex core is homeomorphic to a 3-dimensional manifold with boundary. Geometrically, it is a convex subset of N=H³/G, whose boundary is bent along a geodesic lamination. Assume that the geodesic lamination consists only of a collection A of simple closed curves. We wish to consider the subset of the character variety R(N) consisting of the groups P(A) whose convex cores are bent exactly along A. Our main theorem states that the lengths of the curves in A are global parameters for P(A). In other words, if a₁,...,a_n are the curves in A, the function L : P(A) to Rⁿ of their hyperbolic lengths l(a₁),...,l(a_n) is a homeomorphism onto its image. The main tool used is the theorem of Hodgson-Kerckhoff which gives a local holomorphic parameterization of the deformation space of a hyperbolic cone-manifold.

This should be compared with the work of Bonahon-Otal who first showed the analogous statement with bending angles as parameters instead of lengths. The conjecture is that a group in R(N) is uniquely determined by the bending locus and its bending measure, even in the case that the bending locus is a general geodesic lamination which contains infinite geodesics.