If $\Gamma\subseteq PSL(2,\Bbb R)$ is a geometric finite (i.e. finitely generated; i.e. $\Gamma\backslash\mathbb{H}$ has finite volume) Fuchsian group which is not co-compact and has $\infty$ as a cusp, then why does the minimum \begin{align*} \operatorname{min}\{c>0\mid \begin{pmatrix} *&*\\ c&* \end{pmatrix}\in\Gamma\} \end{align*} exist? I've been reading that this has to do with the "standard polygon" for $\Gamma$ and that $c^{-1}$ is the radius of the largest isometric circle but I don't know the construction of this polygon neither did I find sources of the construction of this. Is the standard polygon connected to the Dirichlet domain (this I know)?
Thanks in advance for any hint.