Fuchsian group

Remarks

The group $\mathrm{PSL}_2(\R)$ acts on the upper half-plane $\mathbb H$ by Möbius transformations $z \mapsto (az + b)/(cz + d)$, and this is the group of orientation-preserving isometries of the hyperbolic metric. A subgroup $\Gamma \subseteq \mathrm{PSL}_2(\R)$ is Fuchsian if and only if its action on $\mathbb H$ is properly discontinuous.

Example: $\mathrm{PSL}_2(\Z)$, the image of the integer matrices of determinant $1$, is Fuchsian. Two distinct integer matrices differ by at least $1$ in some entry, so a small neighbourhood of $\pm I$ contains no other element of the group.

Non-example: the group generated by $z \mapsto z + 1$ and $z \mapsto z + \sqrt2$ consists of the translations $z \mapsto z + a + b\sqrt2$ with $a, b \in \Z$. The translation lengths $a + b\sqrt2$ come arbitrarily close to $0$ without being $0$, so the group is not discrete, and it is not Fuchsian.

Some authors call a group Fuchsian when it is a discrete group of Möbius transformations that preserves a disc or half-plane in the Riemann sphere. Such a group is conjugate to a group as above. Since $\mathrm{PSL}_2(\R) \subset \mathrm{PSL}_2(\C)$, every Fuchsian group is a Kleinian group.