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Nil geometry is one of the eight homogeneous Thurston 3-geometries: E ,S,H,S×R,H×R, S̃L2R, Nil,Sol. Nil can be derived from W. Heisenberg’s famous real matrix group. The notion of translation curve and translation ball can be introduced by initiative of E. Molnár (see [MS], [MSz], [Sz10]). P. Scott in [S] defined Nil lattices to which lattice-like translation ball packings can be defined. In our work we will use the projective model of Nil geometry introduced by E. Molnár in [M97]. In this paper we have studied the translation balls of Nil space and computed their volume. Moreover, we have proved in Theorems 4.1–4.2 that the density of the optimal lattice-like translation ball packing for every natural lattice parameter 1 ≤ k ∈ N is in interval (0.7808, 0.7889) and if r ∈ (0, rd] (rd ≈ 0.7456) then the optimal density is δ Γ ≈ 0.7808. Meanwhile we can apply a nice general estimate of L. Fejes Tóth [LFT] in our Theorem 4.2. From Corollary 4.2 we shall see that the kissing number of the lattice-like ball packings is less than or equal to 14 and the optimal ball packing is realizable in case of equality. We formulate a conjecture for δ Γ , where the density of the conjectural densest packing is δ Γ ≈ 0.7808 for lattice parameter k = 1, larger than the Euclidean one ( π √ 18 ≈ 0.74048), but less than the density of the densest lattice-like geodesic ball packing in Nil space known till now [Sz07]. The kissing number of the translation balls in that packing is 14 as well. Mathematics Subject Classification: 52C17, 52C22, 53A35, 51M20.

Lattice-like translation ball packings in Nil space