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A universal upper bound on density of tube packings in hyperbolic space. (English) Zbl 1098.52005

Based on suitable modifications of techniques which were used for the analogous problem in Euclidean 3-space and are related to Dirichlet domains, the author derives a universal upper bound on the density of packings of tubes (i.e., of sets of points within a fixed distance \(r\) to some line, in each case). He carefully takes into consideration that not all notions referring to Euclidean packing problems have their immediate analogue in the hyperbolic situation (this refers to the choice of definition of a tube, universal upper bound on density instead of precise density, and the dependence of the results on \(r\)). Having these (and further) preparing observations in mind, the author shows that the density of packings of tubes of “radius \(r\)” in hyperbolic 3-space is at most \[ {\sinh r\sin^{-1} {1\over 2\cosh r}\over \sinh^{-1}{\tanh r\over\sqrt{3}}}. \] At the end of the article some applications for hyperbolic 3-manifolds are discussed.

MSC:

52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
51M09 Elementary problems in hyperbolic and elliptic geometries
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