Profinite rigidity of crystallographic groups arising from Lie theory

TL;DR

This paper proves that certain crystallographic groups derived from Lie theory are uniquely determined by their finite quotients, extending previous results on affine Coxeter groups using model theory.

Contribution

It establishes profinite rigidity for products of crystallographic groups from irreducible root systems, generalizing prior work on affine Coxeter groups.

Findings

Profinite rigidity holds for these crystallographic groups.

The proof employs model-theoretic methods.

Results extend understanding of group rigidity in Lie-theoretic contexts.

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.