Editing Fuchsian model

{{Short description|Group representation of a Riemann surface}}
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 geometry|elliptic]], [[parabolic geometry (differential geometry)|parabolic]] or [[hyperbolic geometry|hyperbolic]]. More precisely this theorem states that a Riemann surface <math>R</math> 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]] <math>\mathbb H</math> by a subgroup <math>\Gamma</math> acting [[Properly discontinuous action|properly discontinuously]] and [[Free action|freely]]. 

In the [[Poincaré half-plane model]] for the hyperbolic plane the group of [[biholomorphic transformation]]s is the group <math>\mathrm{PSL}_2(\mathbb R)</math> acting by [[Homography|homographies]], and the uniformization theorem means that there exists a [[Discrete subset|discrete]], [[Torsion-free group|torsion-free]] subgroup <math>\Gamma \subset \mathrm{PSL}_2(\mathbb R)</math> such that the Riemann surface <math>\Gamma \backslash \mathbb H</math> is isomorphic to <math>R</math>. Such a group is called a Fuchsian group, and the isomorphism <math>R \cong \Gamma \backslash \mathbb H</math> is called a Fuchsian model for <math>R</math>.

==Fuchsian models and Teichmüller space ==

Let <math>R</math> be a closed hyperbolic surface and let <math>\Gamma</math> be a Fuchsian group so that <math>\Gamma \backslash \mathbb H</math> is a Fuchsian model for <math>R</math>. Let 
<math display="block">A(\Gamma) = \{ \rho \colon \Gamma \to \mathrm{PSL}_2(\Reals)\colon \rho \text{ is faithful and discrete }\}</math>
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 <math>\Gamma</math> is [[Finitely generated group|finitely generated]] since it is isomorphic to the fundamental group of <math>R</math>. Let <math>g_1, \ldots, g_r</math> be a generating set: then any <math>\rho \in A(\Gamma)</math> is determined by the elements <math>\rho(g_1), \ldots, \rho(g_r)</math> and so we can identify <math>A(\Gamma)</math> with a subset of <math>\mathrm{PSL}_2(\mathbb R)^r</math> by the map <math>\rho \mapsto (\rho(g_1), \ldots, \rho(g_r))</math>. 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: 
{{block indent | em = 1.5 | text = ''For any <math>\rho\in A(\Gamma)</math> there exists a self-[[homeomorphism]] (in fact a [[quasiconformal map]]) <math>h</math> of the upper half-plane <math>\mathbb H</math> such that <math>h \circ \gamma \circ h^{-1} = \rho(\gamma)</math> for all <math>\gamma \in \Gamma</math>.''}}
The proof is very simple: choose an homeomorphism <math>R \to \rho(\Gamma) \backslash \mathbb H</math> and lift it to the hyperbolic plane. Taking a diffeomorphism yields quasi-conformal map since <math>R</math> is compact. 

This result can be seen as the equivalence between two models for [[Teichmüller space]] of <math>R</math>: the set of discrete faithful representations of the fundamental group <math>\pi_1(R)</math> into <math>\mathrm{PSL}_2(\mathbb R)</math> modulo conjugacy and the set of marked Riemann surfaces <math>(X, f)</math> where <math>f\colon R \to X</math> 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).

[[Category:Hyperbolic geometry]]
[[Category:Riemann surfaces]]