Is $PSL_2(\mathbb Z)$ a Fuchsian group of the first kind? We know that as a discrete subgroup of $PSL_2(\mathbb R)$, $PSL_2(\mathbb Z)$ is a Fuchsian group. But how to prove/disprove that it is of the first kind. i.e. if every point on the extended real line is its limit point of some orbit?

Hint: Prove that every rational point on the real line is fixed by a parabolic element of $PSL(2,{\mathbb Z})$. A sub-hint: Think first about stabilizers of nonzero elements of ${\mathbb Z}^2$ in $SL(2,{\mathbb Z})$.  
