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 formula_1 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 formula_2 by a subgroup formula_3 acting properly discontinuously and freely. In the Poincaré half-plane model for the hyperbolic plane the group of biholomorphic transformations is the group formula_4 acting by homographies, and the uniformization theorem means that there exists a discrete, torsion-free subgroup formula_5 such that the Riemann surface formula_6 is isomorphic to formula_1. Such a group is called a Fuchsian group, and the isomorphism formula_8 is called a Fuchsian model for formula_1. Fuchsian models and Teichmüller space. Let formula_1 be a closed hyperbolic surface and let formula_3 be a Fuchsian group so  that formula_6 is a Fuchsian model for formula_1. Let formula_14 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 formula_3 is finitely generated since it is isomorphic to the fundamental group of formula_1. Let formula_17 be a generating set: then any formula_18 is determined by the elements formula_19 and so we can identify formula_20 with a subset of formula_21 by the map formula_22. 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 formula_23 and lift it to the hyperbolic plane. Taking a diffeomorphism yields quasi-conformal map since formula_1 is compact. This result can be seen as the equivalence between two models for Teichmüller space of formula_1: the set of discrete faithful representations of the fundamental group formula_26 into formula_4 modulo conjugacy and the set of marked Riemann surfaces formula_28 where formula_29 is a quasiconformal homeomorphism modulo a natural equivalence relation. References. Matsuzaki, K.; Taniguchi, M.: Hyperbolic manifolds  and Kleinian groups. Oxford (1998).