First-order aspects of Artin groups

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 $n\ge 4$, 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 $K(\ddot{I}€,1)$ conjecture for affine Artin groups.