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Define the following fractional linear transformations (acting on elements of $\mathbb C$):

  • $T_2:\tau \mapsto \tau + 2$
  • $S: \tau \mapsto -1/\tau$

Let $G$ be the group of transformations generated by $T_2$ and $S$.

In my complex analysis textbook, there is a proof that for each point $\tau$ in the upper half-plane, there exists a mapping $g \in G$ such that $g(\tau) \in \mathcal F = \{\tau \in \mathbb C: |\Re(\tau)| \leq 1 \text{ and } \Im(\tau) \geq 0 \text{ and } |\tau| \geq 1 \}$.

Suppose we are given some point $\tau$. The proof begins by choosing $g \in G$ such that $\Im(g(\tau))$ is maximal. My question is, how do we know that there exists such a mapping $g$? (What if, for every $g \in G$, we can find a $g_1 \in G$ such that $\Im(g(\tau)) < \Im(g_1(\tau))$?

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up vote 4 down vote accepted

Note that if $g=\left(\begin{matrix} a & b \\ c & d \end{matrix}\right)$,

$$\Im(g(\tau)) = \frac{\Im \tau}{|c\tau + d|^2}.$$

The numbers $c\tau + d$ form a lattice in $\mathbb{C}$, which is discrete, and therefore $|c\tau + d|$ attains its minimum, which is a positive number. Therefore...

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Thank you! I had not thought about the structure of $G$ at all. – Alan C Jan 20 '12 at 13:31
You are most welcome, Alan! – Bruno Joyal Jan 20 '12 at 18:44

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