Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds

Details Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds Apollonian circle packings and closed horospheres on hyperbolic 3-manifolds

2008-11-14

arxiv.org/abs/0811.2236v5

Mathematics

Dynamical Systems

Final version. To appear in Journal of AMS. Appendix by Oh and Shah

Scientific paper

We obtain an asymptotic formula for the number of circles of curvature at most T in any given bounded Apollonian circle packing. For an integral packing, we obtain the upper bounds for the number of circles with prime curvature as well as of pairs of circles with prime curvatures, which are sharp up constant multiples. The main ingredient of our proof is the effective equidistribution of expanding horospheres on geometrically finite hyperbolic 3-manifolds under the assumption that the critical exponent of its fundamental group exceeds one.

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