In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R as a quotient of the upper half-plane H by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation.
By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. More precisely this theorem states that a Riemann surface R which is not isomorphic to either the Riemann sphere (the elliptic case) or a quotient of the complex plane by a discrete subgroup (the parabolic case) must be a quotient of the hyperbolic plane H by a subgroup Γ acting properly discontinuously and freely.
In the Poincaré half-plane model for the hyperbolic plane the group of biholomorphic transformation is the group P S L 2 ( R ) acting by homographies, and the uniformization theorem means that there exists a discrete, torsion-free subgroup Γ ⊂ P S L 2 ( R ) such that the Riemann surface Γ ∖ H is isomorphic to R . Such a group is called a Fuchsian group, and the isomorphism R ≅ Γ ∖ H is called a Fuchsian model for H .
Fuchsian models and Teichmüller space
Let R be a closed hyperbolic surface and let Γ be a Fuchsian group so that Γ ∖ H is a Fuchsian model for R . Let
A ( Γ ) = { ρ : Γ → P S L 2 ( R ) : ρ is faithful and discrete } and endow this set with the topology of pointwise convergence (sometimes called "algebraic convergence"). In this particular case this topology can most easily be defined as follows: the group Γ is finitely generated since it is isomorphic to the fundamental group of R . Let g 1 , … , g r be a generating set: then any ρ ∈ A ( Γ ) is determined by the elements ρ ( g 1 ) , … , ρ ( g r ) and so we can identify A ( G ) with a subset of P S L 2 ( R ) r by the map ρ ↦ ( ρ ( g 1 ) , … , ρ ( g r ) ) . Then we give it the subspace topology.
The Nielsen isomorphism theorem (this is not standard terminology and this result is not directly related to the Dehn-Nielsen theorem) then has the following statement:
For any ρ ∈ A ( G ) there exists a self-homeomorphism (in fact a quasiconformal map) h of the upper half-plane H such that h ∘ γ ∘ h − 1 = ρ ( γ ) for all γ ∈ G .The proof is very simple: choose an homeomorphism R → ρ ( Γ ) ∖ H and lift it to the hyperbolic plane. Taking a diffeomorphism yields quasi-conformal map since R is compact.
This result can be seen as the equivalence between two models for Teichmüller space of R : the set of discrete faithful representations of the fundamental group π 1 ( R ) into P S L 2 ( R ) modulo conjugacy and the set of marked Riemann surfaces ( X , f ) where f : R → X is a quasiconformal homeomorphism modulo a natural equivalence relation.
