Periodicity and Circle Packing in the Hyperbolic Plane

Details Periodicity and Circle Packing in the Hyperbolic Plane Periodicity and Circle Packing in the Hyperbolic Plane

2003-04-22

arxiv.org/abs/math/0304344v2

Mathematics

Metric Geometry

25 pages

Scientific paper

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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