Fuchsian group

Summary

A Fuchsian group is a discrete group of orientation-preserving isometries of the hyperbolic plane, that is, a discrete subgroup of $\mathrm{PSL}_2(\R)$. It acts properly discontinuously on the upper half-plane, and the quotient is a Riemann surface, possibly with cone points and cusps. By the uniformization theorem, every compact Riemann surface of genus at least $2$ arises as such a quotient. The modular group and its congruence subgroups are the Fuchsian groups behind modular forms.