here is a theorem that the isometries of the Hyperbolic plane are generated by $PSL(2, \mathbb{R})$ and $z \rightarrow - \overline{z}$.

Question

There is a theorem that the isometries of the Hyperbolic plane are generated by $PSL(2, \mathbb{R})$ and $z \rightarrow - \overline{z}$.

My question is, isn't $z \rightarrow kz$ an isometry for $k>0$? I don't see how it is generated by the set in question.

Thanks.

Answer 1

$$ \left( \begin{array}{rr} \sqrt k & 0 \\ 0 & \frac{1}{\sqrt k} \end{array} \right) $$

Answer 2

The scaling map $z \mapsto kz$ for $k > 0$ can be identified with $\begin{pmatrix} \sqrt{k} & 0 \\ 0 & 1/\sqrt{k} \end{pmatrix} \in PSL(2,\mathbb{R})$ because it acts by fractional linear transformation as $$ \begin{pmatrix} \sqrt{k} & 0 \\ 0 & 1/\sqrt{k} \end{pmatrix} z = \frac{\sqrt{k} z + 0}{0 + \frac{1}{\sqrt{k}}} = k z. $$