The weak Bieberbach theorem for crystallographic groups on pseudo-Euclidean spaces

The weak Bieberbach theorem states that each crystallographic group on a Euclidean space uniquely determines its translation lattice as an abstract group. Garipov proved in 2003 that the same holds for crystallographic groups on Minkowski spaces and asked whether a similar claim holds in the pseudo-Euclidean spaces $\mathbb R^{p,q}$. We prove that the weak Bieberbach theorem holds for crystallographic groups on pseudo-Euclidean spaces $\mathbb R^{p,q}$ with $\min\{p,q\}\le2$. For $\min\{p,q\}\ge3$ we construct examples of crystallographic groups with two distinct lattices exchanged by a suitable automorphism of the group. For crystallographic groups with two distinct isomorphic pseudo-Euclidean lattices we also prove that the coranks of their intersection in these lattices can take arbitrary values greater than 2 with the exception of 4.