On Thurston''s uniformization theorem for three-dimensional manifolds

Publisher Summary This chapter presents an outline of Thurston's existence theorem for hyperbolic structures on three-dimensional manifolds-the three-dimensional uniformization theorem. It is based on lectures given by Thurston at the symposium on the Smith conjecture. The chapter provides a complete proof of the uniformization theorem. The criterion for deciding to treat the various pieces was simple: the parts of the argument that are more formal rely on the Bers–Ahlfors–Teichmuller theory or rely on three-dimensional topology. The statements of the main theorem concerning the existence of hyperbolic structures in the case when the manifolds have finite volume are considered. The chapter describes all the topological results necessary to carry out the inductive process of cutting apart the three-manifolds. The chapter focuses on Maskit's combination theorems thatdeal with gluing Kleinian groups together along common quasi-Fuchsian subgroups.