Abstract
Nil geometry is one of the eight 3-dimensional Thurston geometries, it can be derived from W. Heisenberg’s famous real matrix group. The aim of this paper is to study lattice-like ball coverings in Nil space. We introduce the notion of the density of the considered coverings and give upper and lower estimates to it, moreover in Section 3, we formulate a conjecture for the ball arrangement of the least dense latticelike geodesic ball covering and give its covering density \({\triangle \approx 1.42900615}\). The homogeneous 3-spaces have a unified interpretation in the projective 3-sphere and in our work we will use this projective model.
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The research was supported by the grant TÁMOP - 4.2.2.B-10/1, 2010-0009.
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Szirmai, J. On Lattice Coverings of Nil Space by Congruent Geodesic Balls. Mediterr. J. Math. 10, 953–970 (2013). https://doi.org/10.1007/s00009-012-0211-7
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DOI: https://doi.org/10.1007/s00009-012-0211-7