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    Local-global principles in circle packings

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    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 A≀PSL2(K)\mathcal A\leq\textrm{PSL}_2(K) satisfying certain conditions, where KK 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 A\mathcal A possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in PSL2(OK)\textrm{PSL}_2(\mathcal{O}_K) containing a Zariski dense subgroup of PSL2(Z)\textrm{PSL}_2(\mathbb{Z}).Comment: 54 pages, 2 figure
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