Fuchsian model

In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R as a quotient of the upper half-plane H by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation.

By the uniformization theorem, every Riemann surface is either elliptic, parabolic or hyperbolic. More precisely this theorem states that a Riemann surface ${\displaystyle R}$ which is not isomorphic to either the Riemann sphere (the elliptic case) or a quotient of the complex plane by a discrete subgroup (the parabolic case) must be a quotient of the hyperbolic plane ${\displaystyle \mathbb {H} }$ by a subgroup ${\displaystyle \Gamma }$ acting properly discontinuously and freely.

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