Jump to content

Ahlfors finiteness theorem

From Wikipedia, the free encyclopedia
(Redirected from Bers area inequality)

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

[edit]

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

[edit]
  • 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