Slicing, skinning, and grafting

Let M be a compact manifold with boundary. If M is connected, let XC(M) be the SL2(C)–character variety of M. If not, take XC(M) to be the cartesian product of the character varieties of its components. Throughout, S is a closed connected oriented hyperbolic surface, T(S) its Teichmüller space. By the Uniformization Theorem, T(S) is both the space of marked conformal structures on S and the space of marked hyperbolic structures on S; we blur the distinction between these two descriptions, letting context indicate the desired one. The variety XC(S) contains the space AH(S) of hyperbolic structures on S×R. By the work of A. Marden [20] and D. Sullivan [33], the interior of AH(S) is the space of quasifuchsian groups QF(S), and QF(S) lies in the smooth part of XC(S)—though AH(S) sits more naturally in the PSL2(C)–character variety of S, and has many lifts to the variety XC(S), we content ourselves with XC(S), as our arguments apply to any lift considered. We refer the reader to [14] for a detailed treatment of the PSL2(C)– character variety. The quasifuchsian groups are parameterized by the product of Teichmüller spaces T(S)×T(S), by the Simultaneous Uniformization Theorem of L. Bers [2], and the Bers slice BY is the set