Abstract: Recently, there has been a lot of interest in the integers
that appear as curvatures of circles in Apollonian circle packings.
We would first like to survey some of the known theorems about these
integers and their distribution. In particular, there is a famous
theorem of Elena Fuchs that the integer curvatures in an Apollonian
packing satisfy congruence conditions modulo 24, and there is a
conjecture that every sufficiently large integer satisfying those
conditions actually occurs as a curvature.
Then we would like to present analogous theorems about kleinian circle
packings. In particular, all the circles appearing in the limit set of
the 3/10 cusp group in Maskit's parametrization of the deformation
space of once-punctured tori have curvature equal to an integer
multiple of the square root of 11. What is more remarkable is that
every nonnegative integer multiple of the square root of 11 occurs as
a curvature of a circle in the limit set of the 3/10 cusp group.
We hope to present a proof of this.
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