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In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only information about the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface has simply connected universal covering surface given by exactly one of the following:the Riemann sphere,
the complex plane,
the unit disk D or equivalently the upper half-plane H.
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