Sanov subgroup in SL(2,Z)

Definition
This is the subgroup of special linear group:SL(2,Z) generated by the matrices:

$$\begin{pmatrix} 1 & 2 \\ 0 & 1 \\\end{pmatrix}, \qquad \begin{pmatrix} 1 & 0 \\ 2 & 1 \\\end{pmatrix}$$

It is a free group of rank two with the above two elements as a freely generating set for it.

Arithmetic functions
The subgroup has index $$12$$ in the whole group. In fact, any finite index free subgroup of rank two in the special linear group of degree two must have index $$12$$.

Image in projective special linear group
Consider the quotient map $$SL(2,\mathbb{Z}) \to PSL(2,\mathbb{Z})$$. The kernel of this map is of order two. The Sanov subgroup, being free, does not contain any non-identity element of order two, hence it intersects the kernel trivially, so its image in $$PSL(2,\mathbb{Z})$$ is isomorphic to it. By the index considerations, this image is a subgroup isomorphic to free group:F2 of index six inside projective special linear group:PSL(2,Z). For more, see Sanov subgroup in PSL(2,Z).