Lengths are coordinates for convex structures

Abstract

Suppose that N is a geometrically finite orientable hyperbolic 3-manifold. Let P(N,C) be the space of all geometrically finite hyperbolic structures on N whose convex core is bent along a set C of simple closed curves. We prove that the map which associates to each structure in P(N,C) the lengths of the curves in the bending locus C is one-to-one. If C is maximal, the traces of the curves in C are local parameters for the representation space R(N).