On the $Σ$-invariants of Artin groups satisfying the $K(π,1)$-conjecture
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2023
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911780464230400 |
|---|---|
| author | Ferrer, Marcos Escartín Pérez, Conchita Martínez |
| author_facet | Ferrer, Marcos Escartín Pérez, Conchita Martínez |
| contents | We consider $Σ$-invariants of Artin groups that satisfy the $K(π,1)$-conjecture. These invariants determine the cohomological finiteness conditions of subgroups that contain the derived subgroup. We extend a known result for even Artin groups of FC-type, giving a sufficient condition for a character $χ:A_Γ\to\mathbb{R}$ to belong to $Σ^n(A_Γ,\mathbb{Z})$. We also prove some partial converses. As applications, we prove that the $Σ^1$-conjecture holds true when there is a prime $p$ that divides $l(e)/2$ for any edge with even label $l(e)>2$, we generalize to Artin groups the homological version of Bestvina-Brady theorem and we compute the $Σ$-invariants of all irreducible spherical and affine Artin groups and triangle Artin groups, which provide a complete classification of the $F_n$ and $FP_n$ properties of their derived subgroup. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_03091 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the $Σ$-invariants of Artin groups satisfying the $K(π,1)$-conjecture Ferrer, Marcos Escartín Pérez, Conchita Martínez Group Theory Primary 20J06, 20F36, Secondary 57M07, 55P20 We consider $Σ$-invariants of Artin groups that satisfy the $K(π,1)$-conjecture. These invariants determine the cohomological finiteness conditions of subgroups that contain the derived subgroup. We extend a known result for even Artin groups of FC-type, giving a sufficient condition for a character $χ:A_Γ\to\mathbb{R}$ to belong to $Σ^n(A_Γ,\mathbb{Z})$. We also prove some partial converses. As applications, we prove that the $Σ^1$-conjecture holds true when there is a prime $p$ that divides $l(e)/2$ for any edge with even label $l(e)>2$, we generalize to Artin groups the homological version of Bestvina-Brady theorem and we compute the $Σ$-invariants of all irreducible spherical and affine Artin groups and triangle Artin groups, which provide a complete classification of the $F_n$ and $FP_n$ properties of their derived subgroup. |
| title | On the $Σ$-invariants of Artin groups satisfying the $K(π,1)$-conjecture |
| topic | Group Theory Primary 20J06, 20F36, Secondary 57M07, 55P20 |
| url | https://arxiv.org/abs/2309.03091 |