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When $1/p+1/q+1/r<1$, the orientation-preserving hyperbolic triangle group has presentationand is a non-elementary Fuchsian group.

This is the orientation-preserving hyperbolic member of the broader triangle group construction.

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 $\pi /p,\pi /q,\pi /r$ determine the geometry through the sign of $1/p+1/q+1/r-1$. Its orientation-preserving subgroup has group presentation $⟨x,y,z\mid x^p=y^q=z^r=xyz=1⟩$.