Fundamental domain of $\SL(2,\Z)$ (awaiting review)

Let $\mathfrak{F}$ be the following set: \[ \mathfrak{F}=\{z\in\mathcal{H}, |z|\geq1 \text{ and }\Re(z)\leq1/2\}\cup\{\infty\}. \] Then $\mathfrak{F}$ is a fundamental domain for the action of $\Gamma$ on $\mathcal{H}$.