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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) ≤ 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). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ahlfors finiteness theorem」の詳細全文を読む スポンサード リンク
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