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 $\mathcal{A}\le {\text{PSL}}_2(K)$ 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 ${\text{PSL}}_2({\mathcal{O}}_K)$ containing a Zariski dense subgroup of ${\text{PSL}}_2(\mathbb{Z})$.

54 pages, 2 figures

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