In the mathematical theory of

Kleinian group In mathematics, a Kleinian group is a discrete subgroup of the group (mathematics), group of orientation-preserving Isometry, isometries of hyperbolic 3-space . The latter, identifiable with PSL(2,C), , is the quotient group of the 2 by 2 complex ...

s, 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 Ω₁ is an invariant component then :Area(Ω/Γ) ≤ 2Area(Ω₁/Γ) with equality only for

Fuchsian group In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of t ...

s of the first kind (so in particular there can be at most two invariant components).

References

* * * * * Discrete groups Lie groups Kleinian groups Theorems in analysis {{mathematics-stub