Profinite rigidity of affine Coxeter groups

TL;DR

This paper establishes the profinite rigidity of irreducible affine Coxeter groups, demonstrating their first-order rigidity and analyzing their model-theoretic properties, with implications for hyperbolic right-angled Coxeter groups.

Contribution

It proves the profinite rigidity of irreducible affine Coxeter groups and explores their first-order theories and homogeneity properties.

Findings

Affine Coxeter groups are first-order rigid.

Affine Coxeter groups are profinitely rigid.

Hyperbolic right-angled Coxeter groups are homogeneous.

Abstract

We prove that the irreducible affine Coxeter groups are first-order rigid and deduce from this that they are profinitely rigid in the absolute sense. We then show that the first-order theory of any irreducible affine Coxeter group does not have a prime model. Finally, we prove that universal Coxeter groups of finite rank are homogeneous, and that the same applies to every hyperbolic (in the sense of Gromov) one-ended right-angled Coxeter group.