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Rigidity of infinite (circle) packings. (English) Zbl 0726.52008

The following remarkable rigidity theorem for infinite circle packings is proved (the corresponding finite case is already known). Let T be an infinite, planar triangulation, and let Q be a circle packing on the sphere \(S^ 2\) with nerve T. Suppose that the union of all the circular discs of the packing and of the curved triangular regions in the complement of the packing has an at most countable complement. Then every other circle packing on the sphere with nerve T is Möbius equivalent to Q. A special case of this theorem yield the (known) rigidity (i.e., uniqueness up to similarity) of the hexagonal circle packing in the plane. Here the plane is identified with \({\mathbb{C}}\) and \(S^ 2\) with \({\hat {\mathbb{C}}}\); the countable complement in the theorem consists of the point at infinity. The author points out that his proof uses mostly elementary plane topology arguments (no formulae nor inequalities), but that some basic ideas of the theory of quasiconformal maps are hidden in the background.

MSC:

52C15 Packing and covering in \(2\) dimensions (aspects of discrete geometry)
30C62 Quasiconformal mappings in the complex plane
52C25 Rigidity and flexibility of structures (aspects of discrete geometry)
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