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).
Fuchsian groups are used to create Fuchsian models of Riemann surfaces. In some sense, Fuchsian groups do for non-Euclidean geometry what crystallographic groups do for Euclidean geometry. Some Escher graphics are based on them (for the disc model of hyperbolic geometry).
General Fuchsian groups were first studied by Poincaré (1882), who was motivated by the paper (Fuchs 1880) and therefore named them after Lazarus Fuchs.
Read more about Fuchsian Group: Fuchsian Groups On The Upper Half-plane, General Definition, Limit Sets, Examples, Metric Properties
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