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.
A more precise definition
By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. More precisely this theorem states that a Riemann surface 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 by a subgroup acting properly discontinuously and freely.
In the Poincaré half-plane model for the hyperbolic plane the group of biholomorphic transformation is the group acting by homographies, and the uniformization theorem means that there exists a discrete, torsion-free subgroup such that the Riemann surface is isomorphic to . Such a group is called a Fuchsian group, and the isomorphism is called a Fuchsian model for .
Fuchsian models and Teichmüller space
Let be a closed hyperbolic surface and let be a Fuchsian group so that is a Fuchsian model for . Let
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 is finitely generated since it is isomorphic to the fundamental group of . Let be a generating set: then any is determined by the elements and so we can identify with a subset of by the map . 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 there exists a self-homeomorphism (in fact a quasiconformal map) of the upper half-plane such that for all .
The proof is very simple: choose an homeomorphism and lift it to the hyperbolic plane. Taking a diffeomorphism yields quasi-conformal map since is compact.
This result can be seen as the equivalence between two models for Teichmüller space of : the set of discrete faithful representations of the fundamental group into modulo conjugacy and the set of marked Riemann surfaces where is a quasiconformal homeomorphism modulo a natural equivalence relation.
Matsuzaki, K.; Taniguchi, M.: Hyperbolic manifolds and Kleinian groups. Oxford (1998).