Pleating invariants for punctured torus groups

Abstract

In this paper we give a complete description of the space $ \QF $ of quasifuchsian punctured torus groups in terms of what we call {\em pleating invariants}. These are natural invariants of the boundary $\bch$ 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 $\bch$ 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 \cite{MinskyPT}, in which he describes the space of all punctured torus groups in terms of {\em ending invariants} which characterize the asymptotic geometry of the ends of $M$. Pleating invariants give a quasifuchsian analog of the Kerckhoff-Thurston description of Fuchsian space by critical lines and earthquake horocycles. The critical lines extend to {\em pleating planes} on which the pleating loci of $\bch$ are constant and the horocycles extend to {\em BM-slices} on which the pleating invariants of one component of $\bch$ are fixed. We prove that the pleating planes corresponding to rational laminations are dense and that their boundaries can be found {\em 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 $\CC^2$.