Type VF for outer automorphism groups of large-type Artin groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916533860564992 |
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| author | Jones, Oli |
| author_facet | Jones, Oli |
| contents | Given a connected large-type Artin group $A_Γ$, we introduce a deformation space $\mathcal{D}$. If $Γ$ is triangle-free, or has all labels at least 6, we show that this space is canonical, in that it depends only on the isomorphism type of $A_Γ$, and admits an $\Out(A_Γ)$-action. Using this action we conclude that $\Out(A_Γ)$ is of type VF, which implies $\Out(A_Γ)$ finitely presentable. We emphasise that our proof can handle cases where $Γ$ has separating vertices, which were previously problematic.
In fact, our proof works for all connected large-type Artin groups satisfying the technical condition of having rigid chunks. We conjecture that all connected large-type Artin groups have rigid chunks, and therefore outer automorphism groups of type VF. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_17129 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Type VF for outer automorphism groups of large-type Artin groups Jones, Oli Group Theory 20E08, 20F28, 20F36, 20F65 Given a connected large-type Artin group $A_Γ$, we introduce a deformation space $\mathcal{D}$. If $Γ$ is triangle-free, or has all labels at least 6, we show that this space is canonical, in that it depends only on the isomorphism type of $A_Γ$, and admits an $\Out(A_Γ)$-action. Using this action we conclude that $\Out(A_Γ)$ is of type VF, which implies $\Out(A_Γ)$ finitely presentable. We emphasise that our proof can handle cases where $Γ$ has separating vertices, which were previously problematic. In fact, our proof works for all connected large-type Artin groups satisfying the technical condition of having rigid chunks. We conjecture that all connected large-type Artin groups have rigid chunks, and therefore outer automorphism groups of type VF. |
| title | Type VF for outer automorphism groups of large-type Artin groups |
| topic | Group Theory 20E08, 20F28, 20F36, 20F65 |
| url | https://arxiv.org/abs/2410.17129 |