First-order aspects of Artin groups

TL;DR

This paper investigates the model theory of Artin groups, establishing reduction results, characterizing elementary equivalence, and distinguishing affine types using logical and homological methods.

Contribution

It provides new model-theoretic results for Artin groups, including reduction to irreducible components and criteria for elementary equivalence, especially for affine and spherical types.

Findings

Model theory of Artin groups reduces to irreducible components for certain classes.

Elementary equivalence of spherical Artin groups iff they are isomorphic.

Affine Artin groups of type A_n for n  4 are distinguishable via existential sentences.

Abstract

We prove several results on the model theory of Artin groups, focusing on Artin groups which are ``far from right-angled Artin groups''. The first result is that if $\mathcal{C}$ is a class of Artin groups whose irreducible components are acylindrically hyperbolic and torsion-free, then the model theory of Artin groups of type $\mathcal{C}$ reduces to the model theory of its irreducible components. The second result is that the problem of superstability of a given non-abelian Artin group $A$ reduces to certain dihedral parabolic subgroups of $A$ being $n$-pure in $A$, for certain large enough primes $n \in \mathbb{N}$. The third result is that two spherical Artin groups are elementary equivalent if and only if they are isomorphic. Finally, we prove that the affine Artin groups of type $\tilde{A}_n$, for $n \geq 4$, can be distinguished from the other simply laced affine Artin groups using existential sentences; this uses homology results of independent interest relying on the recent proof of the $K(\pi, 1)$ conjecture for affine Artin groups.