Fundamental Polygon of A Compact Riemann Surface
The fundamental polygon of a (hyperbolic) compact Riemann surface has a number of important properties that relate the surface to its Fuchsian model. That is, a hyperbolic compact Riemann surface has the upper half-plane as the universal cover, and can be represented as a quotient manifold H/Γ where Γ is a non-Abelian group isomorphic to the deck transformation group of the surface. The cosets of the quotient space have the standard fundamental polygon as a representative element. In the following, note that all Riemann surfaces are orientable.
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