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Encyclopedia > Fuchsian model

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 π1(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). ...

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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 Rh 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


 
 

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