Fuchsian model
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

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 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 

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: 

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
 the Kleinian model, an analogous construction for 3-manifolds
 Fundamental polygon

References

Matsuzaki, K.; Taniguchi, M.: Hyperbolic manifolds and Kleinian groups. Oxford (1998). Hyperbolic geometry
Riemann surfaces