Title: Local-global principles in circle packings,
Abstract: We generalize work of Bourgain-Kontorovich and Zhang, proving an almost
local-to-global property for the curvatures of certain circle packings, to a
large class of Kleinian groups. Specifically, we associate in a natural way an
infinite family of integral packings of circles to any Kleinian group $\mathcal
A\leq\textrm{PSL}_2(K)$ satisfying certain conditions, where $K$ is an
imaginary quadratic field, and show that the curvatures of the circles in any
such packing satisfy an almost local-to-global principle. A key ingredient in
the proof of this is that $\mathcal A$ possesses a spectral gap property, which
we prove for any infinite-covolume, geometrically finite, Zariski dense
Kleinian group in $\textrm{PSL}_2(\mathcal{O}_K)$ containing a Zariski dense
subgroup of $\textrm{PSL}_2(\mathbb{Z})$.