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
Bers area inequality[edit]
The Bers area inequality is a quantitative refinement of the Ahlfors finiteness theorem proved by Lipman Bers (1967a). It states that if
- 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
- Area(
Ω /Γ ) ≤ 2Area(Ω 1/Γ )
with equality only for Fuchsian groups of the first kind (so in particular there can be at most two invariant components).
References[edit]
- Ahlfors, Lars V. (1964), "Finitely generated Kleinian groups", American Journal of Mathematics, 86: 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: 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: 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: 56–68, doi:10.2307/2373096, ISSN 0002-9327, JSTOR 2373096, MR 0209471