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- Jan 26, 2012

- 995

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**Problem**: Let $G=\text{SL}_2(\mathbb{R})$ and let $\mathcal{H}=\{z\in\mathbb{C}:\text{Im}(z)>0\}$. Let $g=\begin{pmatrix}a & b\\ c & d\end{pmatrix}\in G$ and define an action $\cdot :G\times\mathcal{H}\rightarrow\mathcal{H}$ where $g\cdot z=\dfrac{az+b}{cz+d}$. Compute $\text{Orb}_G(i)$ and $\text{stab}_G(i)$. Is this action transitive?

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