Periodicity and Circle Packing in the Hyperbolic Plane

2003/04/22 by Lewis Bowen, Bowen, Lewis · 3 citations
Mathematics · #52A40 #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG) #math.GR #math.MG #msc:52A40

paper · pdf · doi:10.48550/arxiv.math/0304344

25 pages

arxiv created 2003/09/03 · arxiv updated 2009/11/30

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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