On isomorphisms of crystallographic groups in the pseudo-Euclidean space \(\mathbb{R}^{3,3}\)
V. Churkin
A crystallographic group is a subgroup \(G\) of the motion group of the pseudo-Euclidean space \(\mathbb{R}^{p,q}\) such that the set \(Z\) of all translations in \(G\) is a lattice, i.e. is generated by the translations by vectors of a fixed basis of the space. The factor group \(G/Z\) is isomorphic to a subgroup of the pseudo-orthogonal group \(O_{p,q}(\mathbb{R})\).
In the case of Euclidean spaces, according to Bieberbach's theorem, any isomorphism of crystallographic groups is affine, and an automorphism is induced by a linear transformation of the space. In pseudo-Euclidean spaces, this is not true, in general. If \(\min\{p,q\}\geqslant 3\) then there are crystallographic groups with nonstandard nonlinear automorphisms [1]. These groups contain subgroups that are distinct from the translation lattice, but become translation lattices in other realizations of the group as a crystallographic group. We call these subgroups abstract lattices.
How many abstract lattices can a crystallographic group contain? This problem is solved for \(p=q=3\).
Theorem. A crystallographic group \(G\) in the pseudo-Euclidean space \(\mathbb{R}^{3,3}\) can contain only one, two, or three abstract lattices and no more than two nonstandard automorphisms modulo the subgroup of linear automorphisms.
The subgroup \(N\) of \(G\) generated by all abstract lattices is nilpotent of class one, two, or three. If the upper and lower series of \(N\) coincide then \(N\) is defined uniquely up to isomorphism.
The factor group \(G/N\) is respectively isomorphic to a subgroup of \(O_{p,q}(\mathbb{Z})\), \(SL_3(\mathbb{Z})\), or \(SL_2(\mathbb{Z})\) and may coincide with them.
References
V. A. Churkin, The weak Bieberbach theorem for crystallographic groups on pseudo-Euclidean spaces, Sib. Math. J., 51, N3 (2010), 557–568.
