= A free Kleinian group generated by two parabolic transformations
{c}
{title2=$\langle z\mapsto z+2,\ z\mapsto z/(2z+1)\rangle\cong F_2$}
The integer matrices $A=(\begin{smallmatrix}1&2\\0&1\end{smallmatrix})$ and $B=(\begin{smallmatrix}1&0\\2&1\end{smallmatrix})$ generate a discrete subgroup. For every nonzero integer $m$, $A^m$ sends $|x|<1$ into $|x|>1$, while $B^m$ sends $|x|>1$, including infinity, into $|x|<1$. The <ping-pong lemma> identifies their subgroup as the <free product> of two infinite cyclic groups. This separates the proof of discreteness from the proof of freeness.
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