A **Fuchsian group** is a special type of group in the context of hyperbolic geometry, named after the mathematician Richard Fuchs. More specifically, it is a discrete subgroup of the group of orientation-preserving isometries of the hyperbolic plane, which can be represented as the upper half-plane model \(\mathbb{H}^2\).
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A Fuchsian group is a discrete subgroup of , acting by orientation-preserving isometries of the hyperbolic plane. It is non-elementary when it has no finite orbit in the hyperbolic plane or its ideal boundary.