On the $Σ^1$ and $Σ^2$-invariants of Artin groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913939480117248 |
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| author | Ferrer, Marcos Escartín |
| author_facet | Ferrer, Marcos Escartín |
| contents | We prove the $Σ^1$-conjecture for two families of Artin groups: Artin groups such that there exists a prime number $p$ dividing $\frac{l(e)}{2}$ for every edge $e$ with even label $>2$ and balanced Artin groups. The family of balanced Artin groups extends two previously studied families: the one considered by Kochloukova in arXiv:2009.14269, and the family of coherent Artin groups. We state a conjecture on the $Σ^2$-invariant for Artin groups satisfying the $K(π,1)$-conjecture. The conjecture is proven to be true for two significant families: $2$-dimensional and coherent Artin groups. In the $2$-dimensional case we are able to compute $Σ^n$ for all $n\geq 2$ and to derive finiteness properties of the derived subgroup. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_08692 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the $Σ^1$ and $Σ^2$-invariants of Artin groups Ferrer, Marcos Escartín Group Theory Primary 20J06, 20F36, Secondary 57M07, 55P20 We prove the $Σ^1$-conjecture for two families of Artin groups: Artin groups such that there exists a prime number $p$ dividing $\frac{l(e)}{2}$ for every edge $e$ with even label $>2$ and balanced Artin groups. The family of balanced Artin groups extends two previously studied families: the one considered by Kochloukova in arXiv:2009.14269, and the family of coherent Artin groups. We state a conjecture on the $Σ^2$-invariant for Artin groups satisfying the $K(π,1)$-conjecture. The conjecture is proven to be true for two significant families: $2$-dimensional and coherent Artin groups. In the $2$-dimensional case we are able to compute $Σ^n$ for all $n\geq 2$ and to derive finiteness properties of the derived subgroup. |
| title | On the $Σ^1$ and $Σ^2$-invariants of Artin groups |
| topic | Group Theory Primary 20J06, 20F36, Secondary 57M07, 55P20 |
| url | https://arxiv.org/abs/2501.08692 |