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Homogeneity of irreducible split crystallographic groups

Determine whether every irreducible split crystallographic group is homogeneous in the model-theoretic sense, i.e., whether for all integers n ≥ 1 and all tuples u, v ∈ G^n with the same complete first-order type in G, there exists an automorphism of G mapping u to v.

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Background

The paper proves uniform almost homogeneity (indeed, uniformly almost AE-homogeneity) for irreducible split crystallographic groups and full homogeneity for irreducible affine Coxeter groups. It also provides counterexamples showing that some split crystallographic groups are not homogeneous. The authors highlight a gap: although irreducible split crystallographic groups satisfy strong type-determinacy properties for tuples generating infinite subgroups, homogeneity itself is not established in general.

Clarifying whether irreducible split crystallographic groups are homogeneous would settle an important boundary case in the interaction between model theory and the structure of virtually abelian groups acting on Euclidean spaces.

References

Let us now discuss the irreducibility assumption in Theorem \ref{almost_h}. Notice that the group $G_1\times G_2$ that appears in Theorem \ref{counterexample_theorem} is obviously not irreducible, leading to the following open question.

\begin{qu} Are irreducible (split) crystallographic groups homogeneous? \end{qu}

We do not know the answer to this question (except for irreducible affine Coxeter groups, for which the answer is positive by Theorem \ref{theorem1}), but our next result shows that the failure of homogeneity in a putative non-homogeneous irreducible split crystallographic group cannot be caused by the translation subgroup.

— Homogeneity in Coxeter groups and split crystallographic groups (2504.18354 - André et al., 25 Apr 2025) in Introduction