Fixed subgroups in Artin groups
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866910529592754176 |
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| author | Jones, Oli Vaskou, Nicolas |
| author_facet | Jones, Oli Vaskou, Nicolas |
| contents | We study fixed subgroups of automorphisms of any large-type Artin group $A_Γ$. We define a natural subgroup $\mathrm{Aut}_Γ(A_Γ)$ of $\mathrm{Aut}(A_Γ)$, and for every $γ\in \mathrm{Aut}_Γ(A_Γ)$ we find the isomorphism type of $\mathrm{Fix}(γ)$ and a generating set for a finite index subgroup. We show that $\mathrm{Fix}(γ)$ is a finitely generated Artin group, with a uniform bound on the rank in terms of the number of vertices of $Γ$. Finally, we provide a natural geometric characterisation of the subgroup $\mathrm{Aut}_Γ(A_Γ)$, which informally is the maximal subgroup of $\mathrm{Aut}(A_Γ)$ leaving the Deligne complex of $A_Γ$ invariant. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_11839 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fixed subgroups in Artin groups Jones, Oli Vaskou, Nicolas Group Theory 20F36, 20E36, 20F28, 20F65 We study fixed subgroups of automorphisms of any large-type Artin group $A_Γ$. We define a natural subgroup $\mathrm{Aut}_Γ(A_Γ)$ of $\mathrm{Aut}(A_Γ)$, and for every $γ\in \mathrm{Aut}_Γ(A_Γ)$ we find the isomorphism type of $\mathrm{Fix}(γ)$ and a generating set for a finite index subgroup. We show that $\mathrm{Fix}(γ)$ is a finitely generated Artin group, with a uniform bound on the rank in terms of the number of vertices of $Γ$. Finally, we provide a natural geometric characterisation of the subgroup $\mathrm{Aut}_Γ(A_Γ)$, which informally is the maximal subgroup of $\mathrm{Aut}(A_Γ)$ leaving the Deligne complex of $A_Γ$ invariant. |
| title | Fixed subgroups in Artin groups |
| topic | Group Theory 20F36, 20E36, 20F28, 20F65 |
| url | https://arxiv.org/abs/2407.11839 |