Classification of Sol lattices

TL;DR

This paper classifies $ ext{SOL}$ lattices into 17 types using an algorithmic approach, extending Euclidean lattice classifications, and explores their relations with Minkowskian lattices and $ ext{SOL}$ parallelepipeds.

Contribution

It introduces a novel classification of $ ext{SOL}$ lattices into 17 types and studies their properties and relations with Minkowskian lattices using affine-projective coordinates.

Findings

Classified $ ext{SOL}$ lattices into 17 types.

Established relations between $ ext{SOL}$ and Minkowskian lattices.

Introduced the concept of $ ext{SOL}$ parallelepipeds.

Abstract

$\SOL$ geometry is one of the eight homogeneous Thurston 3-geomet-ri-es $$\EUC, \SPH, \HYP, \SXR, \HXR, \SLR, \NIL, \SOL.$$ In \cite{Sz10} the {\it densest lattice-like translation ball packings} to a type (type {\bf I/1} in this paper) of $\SOL$ lattices has been determined. Some basic concept of $\SOL$ were defined by {\sc{P. Scott}} in \cite{S}, 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 \cite{AQ}. Moreover, we introduce the notion of $\SOL$ parallelepiped to every lattice type. From our new results we emphasize Theorems 3-4-5-6. In this paper we shall use the affine model of $\SOL$ space through affine-projective homogeneous coordinates \cite{M97} which gives a unified way of investigating and visualizing homogeneous spaces, in general.