In mathematics, particularly in the fields of group theory and geometric group theory, a **triangle group** is a type of group that can be defined geometrically by its presentation and associated with the symmetries of triangles in a certain way.
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A triangle group is generated by reflections in the sides of a triangular fundamental domain in spherical geometry, Euclidean geometry or hyperbolic geometry. The angles determine the geometry through the sign of . Its orientation-preserving subgroup has group presentation .