Cyclic surgery, degrees of maps of character curves, and volume rigidity for hyperbolic manifolds

This paper proves the following conjecture of Boyer and Zhang: If a small hyperbolic knot in a homotopy sphere has a non-trivial cyclic surgery slope r, then it has an incompressible surface with non-integer boundary slope strictly between r − 1 and r + 1. I state this result below as Theorem 4.1, after giving the background needed to understand it. Corollary 1.1 of Theorem 4.1 is that any small knot which has only integer boundary slopes has Property P. The proof of Theorem 4.1 also gives information about the diameter of the set of boundary slopes of a hyperbolic knot. The proof of Theorem 4.1 uses a new theorem about the PSL2C character variety of the exterior, M , of the knot. This result, which should be of independent interest, is given below as Theorem 3.1. It says that for certain components of the character variety of M , the map on character varieties induced by ∂M →֒ M is a birational isomorphism onto its image. The proof of Theorem 3.1 depends on a fancy version of Mostow rigidity due to Gromov, Thurston, and Goldman. The connection between Theorem 3.1 and Theorem 4.1 is the techniques introduced by Culler and Shalen which connect the topology of M with its PSL2C character variety. I’ll begin with the background needed for Theorem 4.1. Let K be a knot in a compact, closed, 3-manifold Σ, that is, a tame embedding S →֒ Σ. The exterior of K, M , is Σ minus an open regular neighborhood of K. So M is a compact 3-manifold whose boundary is a torus. Suppose γ is a simple closed curve in ∂M . We can create a closed manifold Mγ from M by taking a solid torus and gluing its boundary to ∂M in such a way that γ bounds a disc in the solid torus (Mγ depends only on the isotopy class of γ). The new manifold Mγ is called a Dehn filling of M or a Dehn surgery on K. Recently,