Local-global principles in circle packings

Abstract

We generalize work by Bourgain and Kontorovich [On the local-global conjecture for integral Apollonian gaskets, Invent. Math.196(2014), 589–650] and Zhang [On the local-global principle for integral Apollonian 3-circle packings, J. Reine Angew. Math.737, (2018), 71–110], 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${\mathcal{A}}\leqslant \text{PSL}_{2}(K)$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 is that${\mathcal{A}}$possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in$\operatorname{PSL}_{2}({\mathcal{O}}_{K})$containing a Zariski dense subgroup of$\operatorname{PSL}_{2}(\mathbb{Z})$.