On Ahlfors' finiteness theorem

Abstract

Ahlfors' finiteness theorem and its two complements, namely the area-inequalities of Bers and the finiteness of the cusps due to Sullivan, are some of the central results in the modern theory of Kleinian groups. Their proofs are analytic whereas their conclusions have a geometric flavor. In this paper we have attempted to explain the topological and group-theoretic genesis of these theorems. Our approach is based on a relative version of the theorem “a finitely generated 3-manifold groups is finitely presented” due to Scott and Shalen.