First-order aspects of Artin groups
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866911081660678144 |
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| author | Cassella, Alberto Paolini, Gianluca Paolini, Giovanni |
| author_facet | Cassella, Alberto Paolini, Gianluca Paolini, Giovanni |
| contents | 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 $K(π, 1)$ conjecture for affine Artin groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21575 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | First-order aspects of Artin groups Cassella, Alberto Paolini, Gianluca Paolini, Giovanni Logic 20F36, 03C60 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 $K(π, 1)$ conjecture for affine Artin groups. |
| title | First-order aspects of Artin groups |
| topic | Logic 20F36, 03C60 |
| url | https://arxiv.org/abs/2507.21575 |