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Arithmetic Fuchsian groups are a special class of Fuchsian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. The prototypical example of an arithmetic Fuchsian group is the modular group P S L 2 {\displaystyle \mathrm {PSL} _{2}}. They, and the hyperbolic surface associated to their action on the hyperbolic plane often exhibit particularly regular behaviour among Fuchsian groups and hyperbolic surfaces.
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