= Solution
A <Kleinian group> is a discrete subgroup of $\operatorname{PSL}_2(\mathbb C)$, acting as orientation-preserving hyperbolic isometries of $\mathbb H^3$ and as Möbius transformations of its boundary sphere. Its limit set consists of the accumulation points on the boundary of an orbit $Gx$ with $x\in\mathbb H^3$; it is independent of that interior base point and is closed. A nonidentity parabolic transformation is conjugate to $z\mapsto z+c$, $c\ne0$. Its iterates on the upper half-space tend to its unique fixed boundary point, infinity in these coordinates. Consequently \b[every parabolic fixed point belongs to the limit set].
The displayed matrix has determinant one, trace two, and is not the identity because $a\ne0$. Its fixed-point equation is $a(z-w)^2=0$, so it has exactly the double fixed point $w$. It is therefore parabolic, and \b[its fixed point is $w$].
The determinant-one Gaussian-integer matrices form a subgroup of $\operatorname{SL}_2(\mathbb C)$. They are discrete because their entries lie in a discrete lattice: a matrix sufficiently near the identity has every entry equal to the corresponding identity entry. Passing through the quotient by $\{\pm I\}$ preserves discreteness, so their Möbius transformations form a Kleinian group.
For $w=n/r$, $n=p+iq$ and $r\ne0$, set $a=r^2$. This gives the <Gaussian-integer parabolic construction>
$$
\boxed{T\ \leftrightarrow\ \begin{pmatrix}1+rn&-n^2\\r^2&1-rn\end{pmatrix}.}
$$
All entries lie in $\mathbb Z[i]$, the determinant is one, and its parabolic fixed point is $n/r=w$. The points with nonzero $p,q,r$ are dense in the finite complex plane, and their closure on the sphere includes infinity. Since the limit set is closed and contains all these parabolic fixed points, \b[the limit set is the whole Riemann sphere].
Back to article page
