I am reading a quick course on modular forms, and I do not understand a lemma: Let $\alpha = \begin{pmatrix}a&b\\c&d \end{pmatrix} \in SL_2(\mathbb{R}), z\in \mathbb{C}^{*}$ (the Riemann sphere), $H$ be the open upper-half complex plane and denote $\alpha(z)=\frac{az+b}{cz+d}$. I am reading a lemma that states:
The map $SL_2(\mathbb{Z})/SO_2(\mathbb{Z}) \rightarrow H$ given by $\alpha \mapsto \alpha(i)$ is a homeomorphism.
I have the following questions:
$1$: Is the domain of the map meant to be $SL_2(\mathbb{Z})$ quotiented by $SO_2(\mathbb{Z})$ or $SL_2(\mathbb{Z})$ set difference $SO_2(\mathbb{Z})$? If the former is correct, can someone describe what the quotient group looks like? If the latter is correct, why do we exclude $SO_2(\mathbb{Z})$ from the domain?
$2$: Can someone give a quick proof that the map is bicontinuous? I have trouble following the proof in the notes.
Thanks in advance!