October 20, 2025Open Access

Profinite rigidity of crystallographic groups arising from Lie theory

Key Points

Every finite direct product of crystallographic groups is profinitely rigid, expanding prior knowledge.
This finding generalizes results on profinite rigidity for affine Coxeter groups, enhancing the framework.
The approach involves model theory, showcasing its applicability in understanding group structures.
These insights highlight the connections between mathematical theories and the rigidity of algebraic structures.

Abstract

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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Cite This Study

Carolillo et al. (Wed,) studied this question.

synapsesocial.com/papers/68f6379bb481a140a36cf7c3 https://doi.org/https://doi.org/10.48550/arxiv.2506.15494

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