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.

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 $PSL(2,\mathbb{Z}) \to PSL(2,\mathbb{R})$.
