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2 Construction and examples
Any hyperbolic metric on a closed, orientable surface of genus is obtained by the following construction: choose a geodesic -gon in the hyperbolic plane with area . (This implies that the sum of interior angles is .) Then choose orientation-preserving isometries which realise the gluing pattern of : for we require that maps to , maps to . Let be the subgroup generated by . Then is a discrete subgroup of and is a hyperbolic surface diffeomorphic to .
The moduli space of hyperbolic metrics on the closed, orientable surface is -dimensional.
5 Further discussion