In Shimura's Intro to Arithmetic Theory of Automorphic Forms, he defines a cusp of a Fuchsian group $\Gamma$ as a point $s \in \mathbb{R} \cup \{ \infty }$ \}$ that is fixed by a parabolic element of $\Gamma$.
I know the definition of the width of a cusp int in the case of $\Gamma = SL_2(\mathbb{Z})$ or a congruence subgroup (namely, it is the only positive integer $h$ such that $\rho^{-1} \overline{\Gamma_s} \rho$ is generated by the matrix $\begin{pmatrix} 1 & h \\ 0 & 1 \end{pmatrix}$, where $\rho$ is any matrix in $SL_2(\mathbb{Z})$ such that $\rho(s) = \infty$) but I was wondering if we can also define a notion of cusp width for any Fuchsian group.
