Fuchsian group (awaiting review)

We say that a subgroup $G$ of $\mathrm {PSL}(2, \mathbb R)$ is a Fuchsian group of the first kind if:

1. It is a discrete subgroup of $\text{PSL}_2(\mathbb{R})$ for the natural topology induced by the topology of $2\times 2$ real matrices
2. 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 $G$.