Ahlfors finiteness theorem
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 , apart from a gap that was filled by . The Ahlfors finiteness theorem states that if Γ is a finitely-generated Kleinian group with region of discontinuity Ω, then Ω/Γ has a finite number of components, each of which is a compact Riemann surface with a finite number of points removed. Bers area inequality. The Bers area inequality is a quantitative refinement of the Ahlfors finiteness theorem  proved by . It states that if Γ is a non-elementary finitely-generated Kleinian group with "N" generators and with region of discontinuity Ω, then with equality only for Schottky groups. (The area is given by the Poincaré metric in each component.) Moreover, if Ω 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).