Gaussian-integer parabolic construction
= Gaussian-integer parabolic construction
{title2=$\begin{pmatrix}1+rn&-n^2\\r^2&1-rn\end{pmatrix}$}
For $n\in\mathbb Z[i]$ and a nonzero integer $r$, the displayed determinant-one matrix has trace two and represents a nonidentity <parabolic Möbius transformation> fixing $n/r$. Its fixed-point equation is $(rz-n)^2=0$. These parabolic fixed points are dense in the <Riemann sphere>, proving that the Gaussian-integer determinant-one <Kleinian group> has full limit set.