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

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 (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 (2): 413–429, doi:10.2307/2373173, ISSN 0002-9327, JSTOR 2373173, MR 0167618
• Ahlfors, Lars (1965), "Correction to "Finitely generated Kleinian groups"", American Journal of Mathematics, 87 (3): 759, doi:10.2307/2373073, ISSN 0002-9327, JSTOR 2373073, MR 0180675
• Bers, Lipman (1967a), "Inequalities for finitely generated Kleinian groups", Journal d'Analyse Mathématique, 18: 23–41, doi:10.1007/BF02798032, ISSN 0021-7670, MR 0229817
• Bers, Lipman (1967b), "On Ahlfors' finiteness theorem", American Journal of Mathematics, 89 (4): 1078–1082, doi:10.2307/2373419, ISSN 0002-9327, JSTOR 2373419, MR 0222282
• Greenberg, L. (1967), "On a theorem of Ahlfors and conjugate subgroups of Kleinian groups", American Journal of Mathematics, 89 (1): 56–68, doi:10.2307/2373096, ISSN 0002-9327, JSTOR 2373096, MR 0209471