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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 Ahlfors (1964, 1965), apart from a gap that was filled by Greenberg (1967).
Ahlfors finiteness theorem

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 Bers (1967a). 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) ≤ 4π(N − 1)

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

Ahlfors, Lars V. (1964), "Finitely generated Kleinian groups", American Journal of Mathematics 86: 413–429, ISSN 0002-9327, JSTOR 2373173, MR0167618
Ahlfors, Lars (1965), "Correction to "Finitely generated Kleinian groups"", American Journal of Mathematics 87: 759, ISSN 0002-9327, JSTOR 2373073, MR0180675
Bers, Lipman (1967a), "Inequalities for finitely generated Kleinian groups", Journal d'Analyse Mathématique 18: 23–41, doi:10.1007/BF02798032, ISSN 0021-7670, MR0229817
Bers, Lipman (1967b), "On Ahlfors' finiteness theorem", American Journal of Mathematics 89: 1078–1082, ISSN 0002-9327, JSTOR 2373419, MR0222282
Greenberg, L. (1967), "On a theorem of Ahlfors and conjugate subgroups of Kleinian groups", American Journal of Mathematics 89: 56–68, ISSN 0002-9327, JSTOR 2373096, MR0209471

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