We give examples of failure of the existence of co-fibered products in the category of algebraic curves.

Hecke groups $H\left({\lambda }_q\right)$ are the discrete subgroups of $\mathrm{P}SL(2,ℝ)$ generated by $S\left(z\right)=-{(z+{\lambda }_q)}^{-1}$ and $T\left(z\right)=-\frac{1}{z}$. The commutator subgroup of $H$(${\lambda }_q)$, denoted by $H^{\text{'}}\left({\lambda }_q\right)$, is studied in [2]. It was shown that $H^{\text{'}}\left({\lambda }_q\right)$ is a free group of rank $q-1$. Here the extended Hecke groups $\overline{H}\left({\lambda }_q\right)$, obtained by adjoining $R_1\left(z\right)=1/\overline{z}$ to the generators of $H\left({\lambda }_q\right)$, are considered. The commutator subgroup of $\overline{H}\left({\lambda }_q\right)$ is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the $H\left({\lambda }_q\right)$ case, the index of $H^{\text{'}}\left({\lambda }_q\right)$ is changed by $q$, in the case of $\overline{H}\left({\lambda }_q\right)$, this number is either 4 for...

We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group $\Gamma$ with cofinite area. As a consequence, we compute the invariants of $\Gamma$, including an explicit finite presentation for $\Gamma$.