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In [[mathematics]], a '''Fuchsian model''' is a construction of a hyperbolic [[Riemann surface]] ''R'' as a quotient of the [[upper half-plane]] '''H'''.  By the [[uniformization theorem]], every Riemann surface is either [[elliptic geometry|elliptic]], [[parabolic geometry|parabolic]] or [[hyperbolic geometry|hyperbolic]]. Every hyperbolic Riemann surface has a non-trivial [[fundamental group]] <math>\pi_1(R)</math>. The fundamental group can be shown to be [[isomorphic]] to some [[group (mathematics)|subgroup]] Γ of the group of real [[Möbius transformation]]s <math>SL(2,\mathbb{R})</math>, this subgroup being a [[Fuchsian group]].  The [[quotient space]] '''H'''/Γ is then a Fuchsian model for the Riemann surface ''R''.  Many authors use the terms ''Fuchsian group'' and ''Fuchsian model'' interchangeably, letting the one stand for the other.  The latter remark is true mostly of the creator of this page.  Meanwhile, Matsuzaki reserves the term Fuchsian model for the Fuchsian group, never the surface itself.

==A more precise definition==
To be more precise, every Riemann surface has a universal [[covering map]] that is either the [[Riemann sphere]], the [[complex plane]] or the [[upper half-plane]]. Given a [[covering map]] <math>f:\mathbb{H}\rightarrow R</math>, where '''H''' is the [[upper half-plane]].

The Fuchsian model of ''R'' is the quotient space <math>R^h = \mathbb{H} / \Gamma</math>. ''R''.  Note that <math>R^h</math> is  a [[complete space|complete]] 2D hyperbolic manifold.

==Nielsen isomorphism theorem==
The Nielsen isomorphism theorem basically states that the [[algebraic topology (object)|algebraic topology]] of a closed Riemann surface is the same as its [[geometry]].

More precisely, let ''R'' be a [[closed]]{{dn|date=July 2012}} hyperbolic surface. Let ''G'' be the Fuchsian group of ''R'' and let <math>\rho:G\rightarrow PSL(2,\mathbb{R})</math> be a faithful representation of ''G'', and let <math>\rho(G)</math> be discrete. Then define the set
:<math>A(G)=\{ \rho : \rho \mbox{ defined as above }\}</math>
and add to this set a [[topological space|topology]] of pointwise convergence, so that ''A''(''G'') is an algebraic topology.  

The '''Nielsen isomorphism theorem''': For any <math>\rho\in A(G)</math> there exists a [[homeomorphism]] ''h'' of the upper half-plane '''H''' such that <math>h \circ \gamma \circ h^{-1} = \rho(\gamma)</math> for all <math>\gamma \in G</math>.

Most of the material here is copied, not very accurately, out of the book below (see page 12).

==References==

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

==See also==
* the [[Kleinian model]], an analogous construction for 3D manifolds
* [[Fundamental polygon]]

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