Profinite rigidity of crystallographic groups arising from Lie theory

2025/06/18 by Davide Carolillo, Carolillo, Davide, Gianluca Paolini +1 · 1 citation
Mathematics · #03C60 #20E18 #20F55 #20H15 #Advanced Algebra and Geometry #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Logic (math.LO)

paper · pdf · doi:10.48550/arxiv.2506.15494

openalex publication_date 2025/06/18 · openalex created_date 2025/10/19 · openalex updated_date 2026/07/28

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.

Citations

• Profinite properties of Coxeter groups
• Homogeneity in Coxeter groups and split crystallographic groups
• On quotients of Coxeter groups
• Distinguishing crystallographic groups by their finite quotients
• Absolute profinite rigidity and hyperbolic geometry
• Automorphisms of Coxeter groups

Cited by

• The profinite genus of the groups ℤⁿ\rtimes C_p²

Related

• Profinite rigidity of affine Coxeter groups
• Profinite rigidity of affine Coxeter groups
• Absolute profinite rigidity, direct products, and finite presentability
• The Profinite Rigidity of Free Metabelian Groups
• First order rigidity of non-uniform higher rank arithmetic lattices