Pleating invariants for punctured torus groups

In this paper we give a complete description of the space QF of quasifuchsian punctured torus groups in terms of what we call pleating invariants. These are natural invariants of the boundary ∂C of the convex core of the associated hyperbolic 3-manifold M and give coordinates for the non-Fuchsian groups QF −F . The pleating invariants of a component of ∂C consist of the projective class of its bending measure, together with the lamination length of a fixed choice of transverse measure in this class. Our description complements that of Minsky in [35], in which he describes the space of all punctured torus groups in terms of ending invariants which characterize the asymptotic geometry of the ends of M . Pleating invariants give a quasifuchsian analog of the KerckhoffThurston description of Fuchsian space by critical lines and earthquake horocycles. The critical lines extend to pleating planes on which the pleating loci of ∂C are constant and the horocycles extend to BM-slices on which the pleating invariants of one component of ∂C are fixed. We prove that the pleating planes corresponding to rational laminations are dense and that their boundaries can be found explicitly. This means, answering questions posed by Bers in the late 1960’s, that it is possible to compute an arbitrarily accurate picture of the shape of any embedding of QF into C.