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 A⩽PSL2(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 A possesses a spectral gap property, which we prove for any infinite-covolume, geometrically finite, Zariski dense Kleinian group in PSL2(OK) containing a Zariski dense subgroup of PSL2(Z) .