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 Lars Ahlfors, apart from a gap that was filled by Greenberg.
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 Lipman Bers. It states that if Γ is a non-elementary finitely-generated Kleinian group with N generators and with region of discontinuity Ω, thenwith equality only for Schottky groups. Moreover, if Ω1 is an invariant component then
with equality only for Fuchsian groups of the first kind.