What does $\overline\Gamma= \Gamma / \lbrace \pm I \rbrace $ mean?

Question

Here $\Gamma $ is a congruence subgroup for $SL_2(\mathbb{Z})$ . The definition says I have to consider $\gamma \cdot (\pm I)$ with $\gamma \in \Gamma$ .

Thanks for help .

Answer 1

When $SL_2(\mathbb{Z})$ acts on the upper half plane, the linear fractional transformation $$\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbb{Z})$$ sends an element $x$ to $$\gamma(x)=\frac{ax+b}{cx+d}.$$ Therefore if you consider the transformation $-I \cdot \gamma$, how does the point $(-I\cdot\gamma)(x)$ compare to the point $\gamma(x)$ for any point $x$?