In mathematics, a **Fuchsian model** is a representation of a hyperbolic Riemann surface *R*. By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. Every hyperbolic Riemann surface has a non-trivial fundamental group π_{1}(*R*). The fundamental group can be shown to be isomorphic to some subgroup Γ of the group of real Mobius transformations , 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. Mathematics, often abbreviated maths in Commonwealth English and math in American English, is the study of abstraction. ...
In mathematics, a group representation is a way of viewing a group in some more concrete way. ...
In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...
In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gauss curvature. ...
In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...
In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ...
In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...
Möbius transformations should not be confused with the Möbius transform. ...
In mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane. ...
For quotient spaces in linear algebra, see quotient space (linear algebra). ...
## 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 , where **H** is the upper half plane... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open...
In mathematics, the Riemann sphere is the unique simply-connected, compact, Riemann surface. ...
The complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. ...
In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ...
In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open...
In mathematics, the upper half plane H is the set of complex numbers x + iy such that y > 0. ...
The Fuchsian model of *R* is the quotient space . *R*. Note that *R*^{h} is a complete 2D hyperbolic manifold. In mathematics and related technical fields, a mathematical object is complete if nothing needs to be added to it. ...
## Nielsen isomorphism theorem The Nielsen isomorphism theorem basically states that the algebraic topology of a closed Riemann surface is the same as its geometry. More precisely, let *R* be a closed hyperbolic surface. Let *G* be the Fuchsian group of *R* and let be a faithful representation of *G*, and let ρ(*G*) be discrete. Then define the set Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...
Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ...
In topology and related branches of mathematics, a closed set is a set whose complement is open. ...
*A*(*G*) = {ρ:ρ defined as above } and add to this set a topology of pointwise convergence, so that *A*(*G*) is an algebraic topology. Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with the study of topological spaces. ...
The **Nielsen isomorphism theorem**: For any there exists a homeomorphism *h* of the upper half-plane **H** such that for all . This word should not be confused with homomorphism. ...
## Related topics An analogous construction for 3D manifolds is the Kleinian model.
## See also |