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 , apart from a gap that was filled by .
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 . 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) ≤
with equality only for
Schottky group
In mathematics, a Schottky group is a special sort of Kleinian group, first studied by .
Definition
Fix some point ''p'' on the Riemann sphere. Each Jordan curve not passing through ''p'' divides the Riemann sphere into two pieces, and we call ...
s. (The area is given by the Poincaré metric in each component.)
Moreover, if Ω
1 is an invariant component then
: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
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Discrete groups
Lie groups
Kleinian groups
Theorems in analysis
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