In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of orientation-preserving 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) (so that it contains orientation-reversing elements), and sometimes it is allowed to be a Kleinian group (a discrete subgroup of PSL(2,C)) which is conjugate to a subgroup of PSL(2,R).

Fuchsian groups are used to create Fuchsian models of Riemann surfaces. In this case, the group may be called the Fuchsian group of the surface. 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 Henri Poincaré (1882), who was motivated by the paper (Fuchs 1880), and therefore named them after Lazarus Fuchs.

Let ${\displaystyle H=\{z\in \mathbb {C} |\operatorname {Im} {z}>0\}}$ be the upper half-plane. Then ${\displaystyle H}$ is a model of the hyperbolic plane when endowed with the metric

The group PSL(2,R) acts on ${\displaystyle H}$ by linear fractional transformations (also known as Möbius transformations):

This action is faithful, and in fact PSL(2,R) is isomorphic to the group of all orientation-preserving isometries of ${\displaystyle H}$.

A Fuchsian group ${\displaystyle \Gamma }$ may be defined to be a subgroup of PSL(2,R), which acts discontinuously on ${\displaystyle H}$. That is,

An equivalent definition for ${\displaystyle \Gamma }$ to be Fuchsian is that ${\displaystyle \Gamma }$ be a discrete group, which means that: