| Abstract: In this talk, we would like to go through the details of the circle
packing which is the limit set of the Kleinian group
$G=\langle a,b\rangle$ generated by:
\[ a =\begin{bmatrix}
\frac{\sqrt{11}-i}{2} & -i\\
-i&0
\end{bmatrix}
,\quad
b= \begin{bmatrix}
1&2\\0&1
\end{bmatrix}.
\]
Previously we have identified this group as the $3/10$ cusp group on
the boundary of Maskit's embedding of the Teichmüller space of
once-punctured tori. We have also explained that this circle packing
has unusual number-theoretic properties. The circle packing consists
of the disjoint disk orbits $G(D_0)$ and $G(D_1)$ for the lower
half-plane $D_0$ and the disk $D_1$ of radius $\frac{1}{\sqrt{11}}$
which is externally tangent to $D_0$ at $-1$. These correspond to the
normalized Hermitian matrices:
\[
H_0
=\left[\begin{array}{cc}0 & i \\ -i & 0 \end{array}\right], \quad
H_1
=\left[\begin{array}{cc}\sqrt{11} & \sqrt{11}-i \\ \sqrt{11}+i & \sqrt{11}
\end{array}\right].
\]
Since the image of a disk with Hermitian matrix $H$ under the group
element $g\in G$ corresponds to the Hermitian matrix $g(H) =
{g^*}^{-1} H g^{-1}$, we have shown by direct calculation that all
circles in the packing have Hermitian matrices of the form
$
H=\begin{bmatrix}
s\sqrt{11} & \frac{m\sqrt{11} +n\,i}2\\
\frac{m\sqrt{11}
-n\,i}{2} & t\sqrt{11}
\end{bmatrix}$, for nonnegative integers $s,t$ and
integers $m,n$ satisfying $m\equiv n\pmod{2}$ and determinant
$11st-\frac{11m^2+n^2}{4}=-1$. Thus, all circle curvatures are
$s\sqrt{11}$ for nonnegative integers $s$. In this talk, we wish to describe in more detail the linear and
quadratic relations satisfied by finite chains of circles in this
packing. We would also like to explain the resemblance of this
packing to the Apollonian packing in the following sense. An
Apollonian packing begins with three mutually externally tangent disks
$D_1$, $D_2$, $D_3$ and the two disks $D_4$, $D_4'$ which are
externally tangent to the original three disk chain. In the 3/10
circle packing, there are 5 disjoint disks $D_1$, $D_2$, $D_3$, $D_4$,
$D_4'$ which are connected by single disks in between certain
pairs. Thus, the packing can be viewed as shrinking the original 5
disks in the Apollonian packing and inserting single disks at places where
tangent points were previously located. The picture will make this clearer. |