In the mathematical theory of Kleinian groups, the Ahlfors finiteness theorem describes the quotient of the domain of discontinuity by a finitely generated Kleinian group. The theorem was proved by Lars Ahlfors, apart from a gap that was filled by Greenberg.
The Ahlfors finiteness theorem states that if àis a finitely-generated Kleinian group with region of discontinuity é, then é/àhas a finite number of components, each of which is a compact Riemann surface with a finite number of points removed.
The Bers area inequality is a quantitative refinement of the Ahlfors finiteness theorem proved by Lipman Bers. It states that if àis a non-elementary finitely-generated Kleinian group with N generators and with region of discontinuity é, then
with equality only for Schottky groups. (The area is given by the Poincaré metric in each component.) Moreover, if é<sub>1</sub> is an invariant component then
with equality only for Fuchsian groups of the first kind (so in particular there can be at most two invariant components).