Theorems show profinite rigidity in crystallographic groups, suggesting significant implications for root systems.
We prove that every finite direct product of crystallographic groups arising from an irreducible root system (in the sense of Lie theory) is profinitely rigid (equiv. first-order rigid). This is a generalization of recent proofs of profinite rigidity of affine Coxeter groups [1, 7, 22]. Our proof uses model theory.
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Carolillo et al. (2025) studied this question.
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