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

Abstract

This paper proves a theorem about Dehn surgery using a new theorem about PSL(2, C) character varieties. Confirming a conjecture of Boyer and Zhang, this paper shows that a small hyperbolic knot in a homotopy sphere having a non-trivial cyclic slope r has an incompressible surface with non-integer boundary slope strictly between r-1 and r+1. A corollary is that any small knot which has only integer boundary slopes has Property P. The proof uses connections between the topology of the complement of the knot, M, and the PSL(2, C) character variety of M that were discovered by Culler and Shalen. The key lemma, which should be of independent interest, is that for certain components of the character variety of M, the map on character varieties induced by the inclusion of boundary M into M is a birational isomorphism onto its image. This in turn depends on a fancy version of Mostow rigidity due to Gromov, Thurston, and Goldman.