Density in hyperbolic spaces
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We present new (and old) examples showing the difficulty of defining density for packings of hyperbolic space. Using probabilistic techniques, we develop a new method for studying density and show that it corresponds to well-founded notions in Euclidean space. Using this new machinery, we prove a conjecture of G. Fejes Toth, G. Kuperberg and W. Kuperberg regarding the existence of locally densest packings. We also prove that for most radii r all optimally dense packings of hyperbolic space by spheres of radius r have low symmetry.