Chasing maximal pro-p Galois groups via 1-cyclotomicity
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arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866929265052745728 |
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| author | Quadrelli, Claudio |
| author_facet | Quadrelli, Claudio |
| contents | Let $p$ be a prime. We prove that certain amalgamated free pro-$p$ products of Demushkin groups with pro-$p$-cyclic amalgam cannot give rise to a 1-cyclotomic oriented pro-$p$ group, and thus do not occur as maximal pro-$p$ Galois groups of fields containing a root of 1 of order $p$. We show that other cohomological obstructions which are used to detect pro-$p$ groups that are not maximal pro-$p$ Galois groups - the quadraticity of $\mathbb{Z}/p\mathbb{Z}$-cohomology and the vanishing of Massey products - fail with the above pro-$p$ groups. Finally, we prove that the Minač-Tân pro-$p$ group cannot give rise to a 1-cyclotomic oriented pro-$p$ group, and we conjecture that every 1-cyclotomic oriented pro-$p$ group satisfy the strong $n$-Massey vanishing property for $n>2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_00335 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Chasing maximal pro-p Galois groups via 1-cyclotomicity Quadrelli, Claudio Group Theory Primary 12G05, Secondary 20E18, 20J06, 12F10 Let $p$ be a prime. We prove that certain amalgamated free pro-$p$ products of Demushkin groups with pro-$p$-cyclic amalgam cannot give rise to a 1-cyclotomic oriented pro-$p$ group, and thus do not occur as maximal pro-$p$ Galois groups of fields containing a root of 1 of order $p$. We show that other cohomological obstructions which are used to detect pro-$p$ groups that are not maximal pro-$p$ Galois groups - the quadraticity of $\mathbb{Z}/p\mathbb{Z}$-cohomology and the vanishing of Massey products - fail with the above pro-$p$ groups. Finally, we prove that the Minač-Tân pro-$p$ group cannot give rise to a 1-cyclotomic oriented pro-$p$ group, and we conjecture that every 1-cyclotomic oriented pro-$p$ group satisfy the strong $n$-Massey vanishing property for $n>2$. |
| title | Chasing maximal pro-p Galois groups via 1-cyclotomicity |
| topic | Group Theory Primary 12G05, Secondary 20E18, 20J06, 12F10 |
| url | https://arxiv.org/abs/2106.00335 |