The Grafting Map of Teichmüller Space

1.1 Statement and Context. One of the underlying principles in the study of Kleinian groups is that aspects of the complex projective geometry of quotients of Ĉ by the groups reflect properties of the threedimensional hyperbolic geometry of the quotients of H by the groups. Yet, even though it has been over thirty-five years since Lipman Bers wrote down a holomorphic embedding of the Teichmüller space of Riemann surfaces in terms of the projective geometry of a Teichmüller space of quasi-Fuchsian manifolds, no corresponding parametrization in terms of the three-dimensional hyperbolic geometry has been presented. One of the goals of this paper is to give such a parametrization. This parametrization is straightforward and has been expected for some time ([Ta97], [Mc98]): to each member of a Bers slice of the space QF of quasi-Fuchsian 3-manifolds, we associate the bending measured lamination of the convex hull facing the fixed “conformal” end. The geometric relationship between a boundary component of a convex hull and the projective surface at infinity for its end is given by a process known as grafting, an operation on projective structures on surfaces that traces its roots back at least to Klein [Kl33;§50, p. 230], with a modern history developed by many authors ([Ma69], [He75], [Fa83], [ST83], [Go87],[GKM95],[Ta97],[Mc98]). The main technical tool in our proof that bending measures give coordinates for Bers slices, and the second major goal of this paper, is the completion of the proof of the “Grafting Conjecture”. This conjecture states that for a fixed measured lamination λ, the self-map of Teichmüller space induced by grafting a surface along λ is a homeomorphism of Teichmüller space; our contribution to this argument is a proof of the injectivity of the grafting map. While the principal application