Local-global Principles in Circle Packings
نویسندگان
چکیده
We generalize work of Bourgain-Kontorovich [6] and Zhang [31], 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 of this 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).
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