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. The concept is named after Lazarus Fuchs.
A more precise definition
editBy 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 transformations 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
editLet be a closed hyperbolic surface and let be a Fuchsian group so that is a Fuchsian model for . Let
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:
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.
See also
edit- the Kleinian model, an analogous construction for 3-manifolds
- Fundamental polygon
References
editMatsuzaki, K.; Taniguchi, M.: Hyperbolic manifolds and Kleinian groups. Oxford (1998).