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Classification of \(\mathrm{Sol}\) lattices. (English) Zbl 1262.51005

Solvegeometry denotes a geometry modelled on the \(3\)-dimensional Lie group \(\mathrm{Sol}\) which is diffeomorphic to \(\mathbb{R}^3\) with group product \[ (x,y,z)\cdot(x',y',z')=(x'+x\mathrm{e}^{-z'},y'+y\mathrm{e}^{z'},z+z'). \] The actions by right-multiplication of the group \(\mathrm{Sol}\) on the space \(\mathrm{Sol}\) are called the translations of \(\mathrm{Sol}\) by the authors.
A \(\mathrm{Sol}\)-lattice is a discrete subgroup \(\Gamma\) of \(\mathrm{Sol}\) acting cocompactly on \(\mathrm{Sol}\) and generated by three translations \(\tau_1,\tau_2,\tau_3\).
The authors give a classification of the \(\mathrm{Sol}\)-lattices analogous to the Bravais classification of lattices in Euclidean space. This classification is based on four step decision procedure; first, the lattices are seperated into two main types I and II, depending on whether the translation \(\tau_3\) has a component acting on the \(xy\)-plane (II) or not (I), where \(\tau_1\) and \(\tau_2\) are assumed to act in the \(xy\)-plane. In the next step, the lattices are further distinguished by their point groups, those isometrie of \(\mathrm{Sol}\) fixing the origin and mapping \(\Gamma\) to itself. The third and fourth step apply to main type II only and concern the relations between the \(x\)- and \(y\)-component of \(\tau_3\) to its \(z\)-component.
As a side note let it be said that the journal’s editors did not do a good job; the paper contains quite a few typos and the authors’ language is somewhat confusing in many places. The editors should have taken care to remedy that.

MSC:

51D99 Geometric closure systems
52C17 Packing and covering in \(n\) dimensions (aspects of discrete geometry)
52C07 Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry)
57S30 Discontinuous groups of transformations
22E25 Nilpotent and solvable Lie groups
22E40 Discrete subgroups of Lie groups
57M60 Group actions on manifolds and cell complexes in low dimensions
53A35 Non-Euclidean differential geometry
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References:

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