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,[1][2] apart from a gap that was filled by Greenberg.[3]
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.[4] 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. (April 1964). "Finitely Generated Kleinian Groups". American Journal of Mathematics 86 (2): 413. doi:10.2307/2373173. https://www.jstor.org/stable/2373173?origin=crossref.
- ↑ "Errata". American Journal of Mathematics 7 (1). October 1884. doi:10.2307/2369455. ISSN 0002-9327. https://doi.org/10.2307/2369455.
- ↑ Greenberg, L. (January 1967). "On a Theorem of Ahlfors and Conjugate Subgroups of Kleinian Groups". American Journal of Mathematics 89 (1): 56. doi:10.2307/2373096. https://www.jstor.org/stable/2373096?origin=crossref.
- ↑ Bers, Lipman (December 1967). "Inequalities for finitely generated Kleinian groups" (in en). Journal d'Analyse Mathématique 18 (1): 23–41. doi:10.1007/BF02798032. ISSN 0021-7670. http://link.springer.com/10.1007/BF02798032.
Further reading
- Bers, Lipman (1967b), "On Ahlfors' finiteness theorem", American Journal of Mathematics 89 (4): 1078–1082, doi:10.2307/2373419, ISSN 0002-9327
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