Fuchsian group (awaiting review)

A Fuchsian group is a discrete subgroup of $\mathrm{PSL}(2,\mathbb{R})$ for the natural topology induced by the topology of $2\times 2$ real matrices.

We say that a Fuchsian group is of the first kind if it has finite covolume for the hyperbolic metric $d\mu=dxdy/y^2$, i.e. \[ \text{covol}(G)=\int_{\mathfrak{F}(G)}d\mu<\infty, \] where $\mathfrak{F}(G)$ is the fundamental domain of $G$.