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Let $\Gamma(2)$ be the subgroup of $\mathrm{SL}_2(\mathbf{Z})$ satisfying the usual congruence conditions. It acts on the complex upper half-plane.

Does it have a fundamental domain contained in the strip $$\{x+iy: -1\leq x \leq 1\}?$$

Is the following a correct argument?

The matrix $$\left( \begin{matrix} 1 & \pm 2 \\ 0 & 1 \end{matrix}\right)$$ is in $\Gamma(2)$. If $\tau$ is not in the above strip, we can translate $\tau$ into this strip by multiplying with the above matrix a couple of times.

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    the argument is certainly valid. If you want a particular fund. domain in your strip then your strip minus the two half-discs with centers $\pm1/2$ and radii $1/2$ is an example.2012-04-17
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    Ow that reminds me a lot of the standard fundamental domain of $\mathrm{SL}_2(\mathbf{Z})$.2012-04-17

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