First-order aspects of Artin groups

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 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 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∈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 A~n, for n⩾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​(π,1) conjecture for affine Artin groups.