Arithmetic Fuchsian group

Summary

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 $\mathrm{PSL}_2(\mathbb{Z} )$. They, and the hyperbolic surface associated to their action on the hyperbolic plane, often exhibit particularly regular behaviour among Fuchsian groups and hyperbolic surfaces.