Abstract
We prove that given a fixed radius r, the set of isometry-invariant probability measures supported on 'periodic' radius r-circle packings of the hyperbolic plane is dense in the space of all isometry-invariant probability measures on the space of radius r-circle packings. By a periodic packing, we mean one with cofinite symmetry group. As a corollary, we prove the maximum density achieved by isometry-invariant probability measures on a space of radius r-packings of the hyperbolic plane is the supremum of densities of periodic packings. We also show that the maximum density function varies continuously with radius.
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Bowen, L. Periodicity and Circle Packings of the Hyperbolic Plane. Geometriae Dedicata 102, 213–236 (2003). https://doi.org/10.1023/B:GEOM.0000006580.47816.e9
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DOI: https://doi.org/10.1023/B:GEOM.0000006580.47816.e9