A proof of the Ahlfors finiteness theorem

Summary

Introduction

The modern theory of hyperbolic 3-manifolds began with the Ahlfors Finiteness Theorem. It states that the quotient Ω(G)/G of the ordinary set of a finitely generated Kleinian group G is a finite union of surfaces, each of which is a closed surface with at most a finite number of punctures and cone points. We will formally state Ahlfors' theorem in §4. His 1964 proof [Ahl66] involved delicate analytic estimates for automorphic forms. In his enthusiasm, he forgot to rule out the possibility of infinitely many triply punctured spheres (or triangle groups more generally). The omission was soon rectified both by Bers [Ber67] with further analysis of the forms, and more elegantly by Greenberg [Gre67] using a lemma akin to Selberg's lemma. See [Gre77] and [Kra72] for expositions of these proofs. As the theory of hyperbolic manifolds developed around 1970 it was recognized that much of the proof can be carried out more simply and naturally using topological considerations [Mar74]. The topological approach was completed when the theory of the compact core and relative core was brought in, first by Kulkarni–Shalen [KS89], and then in its full implementation by Feighn–McCullough [FM87]. What then remains to do for the proof of Ahlfors' theorem is to rule out boundary components of the 3-manifold being topological disks or more generally bordered Riemann surfaces.

In the wake of Sullivan's proof of the no wandering domain theorem [Sul85] the whole shebang, especially the analytic part, became greatly simplified.