Lengths are coordinates for convex structures

Abstract

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