Group structure of $\Gamma_{0}(2)$

Question

It is known that the (projective) modular group $\mathrm{PSL}_{2}(\mathbb{Z})$ is generated by $$T=\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}, S=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$$ with $S^{2}=(ST)^{3}=I$. So the group is isomorphic to free product of two cyclic group, $\mathbb{Z}/2\mathbb{Z}\ast \mathbb{Z}/3\mathbb{Z}$. Is there any similar characterization for congruence subgroups, $\Gamma(N), \Gamma_{0}(N)$ and $\Gamma_{1}(N)$? Especially, I want to know structure of $\Gamma_{0}(2)$, which is generated by $$T=\begin{pmatrix} 1 & 1 \\ 0 & 1\end{pmatrix}, R=\begin{pmatrix} 1 & 0 \\ 2 & 1\end{pmatrix}=ST^{-2}S$$ Actually, it seems that there aren't any relation between $T$ and $R$, so I think that $\Gamma_{0}(2)$ is isomorphic to free group $F_{2}$. Is it true?

Answer 1

There do exist relations between these elements. If $T$ and $R$ are your generators, then $R T^{-1} = \left(\begin{array}{rr} 1 & -1 \\ 2 & -1 \end{array}\right)$ and hence $(RT^{-1})^2$ is the identity.

This is actually the only relation, so your group $\Gamma_0(2)$ is in fact isomorphic to a free product of $\mathbf{Z}$ and $\mathbf{Z} / 2$. To prove this you can use Farey symbols, which are a general tool for computing presentations of any finite-index subgroup of $PSL_2(\mathbf{Z})$ (see Kurth + Long 2007 for a survey).