Fuchsian group

In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of the upper half plane, so a Fuchsian group can be regarded as a group acting on any of these spaces. There are some variations of the definition: sometimes the Fuchsian group is assumed to be finitely generated, sometimes it is allowed to be a subgroup of PGL(2,R) = PSL(2,R).2 (so that it contains orientation-reversing elements) and sometimes it is allowed to be a Kleinian group (a discrete group of PSL(2,C)) that is conjugate to a subgroup of PSL(2,R).