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, then :Area(Ω/Γ) ≤ with equality only for
Schottky groups. (The area is given by the Poincaré metric in each component.) Moreover, if Ω1 is an invariant component then :Area(Ω/Γ) ≤ 2Area(Ω1/Γ) with equality only for
Fuchsian groups of the first kind (so in particular there can be at most two invariant components). ==References==