Fuchsian group - Knowlpedia

A Fuchsian group is a discrete subgroup $\Gamma\le\operatorname{PSL}_2(\mathbb R)$ , with the usual Lie-group topology. It acts on the upper half-plane $\mathcal H=\{z:\operatorname{Im}z>0\}$ by

\begin{pmatrix}a&b\\c&d\end{pmatrix}\!\cdot z=\frac{az+b}{cz+d}.

Matrices differing by sign give the same orientation-preserving hyperbolic isometry.

Scope

Finite covolume, torsion-freeness, and arithmeticity are extra conditions. The definition here includes elementary and finite groups.

For example, the group generated by $z\mapsto z+1$ is Fuchsian. Its quotient has infinite hyperbolic area: on a vertical strip of width $1$ , the area integral $\int_0^1\int_0^1 y^{-2}\,dy\,dx$ diverges. Thus discreteness alone does not imply finite covolume.

References

John Voight, Quaternion Algebras, §34.7, Definition 34.7.3.