the presentation of $SL(2,\mathbb{Z})$ There is a natural presentation $SL(2,\mathbb{Z})\hookrightarrow GL(2,\mathbb{R})$, are there other presentations in real dimension 2? Or there is a classification of all the presentation of $SL(2,\mathbb{Z})\to GL(2,\mathbb{R})$? Thanks in advance.
 A: J.P. Serre, Trees, Springer-Verlag (1980), p. 81,
$$\mathrm{SL}_2(\mathbb{Z}) = \langle \,S, T \mid S^4 = 1, (ST)^3 = S^2 \,\rangle$$
with,
\begin{align}
    S &=
    \begin{pmatrix}
    \phantom{-}0& 1 \\ -1 & 0
    \end{pmatrix},
    &
    T &= 
    \begin{pmatrix}
    1 & 0 \\ 1 & 1
    \end{pmatrix}.
\end{align}
A: The classification is known. It is a bit easier to work with the projetive modulear group $PSL(2,\mathbb{Z})=SL(2,\mathbb{Z})/\pm I$. This group is isomorphic to the free product of $\mathbb{Z}/2\ast \mathbb{Z}/3$, and finite-dimensional representations of $PSL(2,\mathbb{Z})$ correspond bijectively to finite-dimensional modules of the group algebra $k[PSL(2,\mathbb{Z})]=k[\mathbb{Z}/2\ast \mathbb{Z}/3]\simeq k[\mathbb{Z}/2]\ast k[\mathbb{Z}/3]\simeq k\langle x,y,\rangle /( x^3-1,y^2-1)$. The $2$-dimensional modules are classified here, in section $1.3$.
A: To supplement the answer of @Dietrich Burde, the representations $PSL(2,\mathbb{Z}) = \mathbb{Z}/2 * \mathbb{Z}/3 \to PSL(2,\mathbb{R})$ correspond bijectively to ordered pairs of elements $X,Y \in PSL(2,\mathbb{R})$ such that $X$ has order $1$ or $2$ and $Y$ has order $1$ or $3$; equivalently, $X$ is the identity or has trace $0$, and $Y$ is the identity or has trace $\pm 1$. 
It is also interesting that amongst all such representations, the ones which are discrete, faithful, and have the same parabolics as $PSL(2,\mathbb{Z})$ are precisely the ones which are conjugate to the inclusion $PSL(2,\mathbb{Z}) \to PSL(2,\mathbb{R})$.
