Fuchsian Model

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, parabolic or hyperbolic. Every hyperbolic Riemann surface has a non-trivial fundamental group . The fundamental group can be shown to be isomorphic to some subgroup Γ of the group of real Möbius 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. 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.