Classification of Sol lattices

Abstract

Sol geometry is one of the eight homogeneous Thurston 3-geometries

$${\bf E}^{3}, {\bf S}^{3}, {\bf H}^{3}, {\bf S}^{2}\times{\bf R}, {\bf H}^{2}\times{\bf R}, \widetilde{{\bf SL}_{2}{\bf R}}, {\bf Nil}, {\bf Sol}.$$

In [13] the densest lattice-like translation ball packings to a type (type I/1 in this paper) of Sol lattices has been determined. Some basic concept of Sol were defined by Scott in [10], in general. In our present work we shall classify Sol lattices in an algorithmic way into 17 (seventeen) types, in analogy of the 14 Bravais types of the Euclidean 3-lattices, but infinitely many Sol affine equivalence classes, in each type. Then the discrete isometry groups of compact fundamental domain (crystallographic groups) can also be classified into infinitely many classes but finitely many types, left to other publication. To this we shall study relations between Sol lattices and lattices of the pseudoeuclidean (or here rather called Minkowskian) plane [1]. Moreover, we introduce the notion of Sol parallelepiped to every lattice type. From our new results we emphasize Theorems 3–6. In this paper we shall use the affine model of Sol space through affine-projective homogeneous coordinates [6] which gives a unified way of investigating and visualizing homogeneous spaces, in general.