Sylvester domains and pro-$p$ groups
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915811369680896 |
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| author | Jaikin-Zapirain, Andrei Souza, Henrique |
| author_facet | Jaikin-Zapirain, Andrei Souza, Henrique |
| contents | Let $G$ be a finitely generated torsion-free pro-$p$ group containing an open free-by-$\mathbb{Z}_p$ pro-$p$ subgroup. We show that the completed group algebra of $G$ over $\mathbb{F}_p$ is a Sylvester domain. Moreover the inner rank of a matrix $A$ over this completed group algebra can be calculated by approximation by ranks corresponding to finite quotients of $G$, that is, if $G=G_1>G_2>\ldots$ is a chain of normal open subgroups of $G$ with trivial intersection and $A_i$ is the matrix over $\mathbb{F}_p[G/G_i]$ obtained from the matrix $A$ by applying the natural homomorphism induced from $G \to G/G_i$, then the inner rank of $A$ equals $\lim_{i\to \infty} \frac{\operatorname{rk}_{\mathbb{F}_p} (A_i)}{|G:G_i|}$. As a consequence, we obtain a particular case of the mod $p$ Lück approximation for abstract finitely generated subgroups of free-by-$\mathbb{Z}_p$ pro-$p$ groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14130 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sylvester domains and pro-$p$ groups Jaikin-Zapirain, Andrei Souza, Henrique Group Theory Rings and Algebras 20E18, 20E22, 16S35, 16S85 Let $G$ be a finitely generated torsion-free pro-$p$ group containing an open free-by-$\mathbb{Z}_p$ pro-$p$ subgroup. We show that the completed group algebra of $G$ over $\mathbb{F}_p$ is a Sylvester domain. Moreover the inner rank of a matrix $A$ over this completed group algebra can be calculated by approximation by ranks corresponding to finite quotients of $G$, that is, if $G=G_1>G_2>\ldots$ is a chain of normal open subgroups of $G$ with trivial intersection and $A_i$ is the matrix over $\mathbb{F}_p[G/G_i]$ obtained from the matrix $A$ by applying the natural homomorphism induced from $G \to G/G_i$, then the inner rank of $A$ equals $\lim_{i\to \infty} \frac{\operatorname{rk}_{\mathbb{F}_p} (A_i)}{|G:G_i|}$. As a consequence, we obtain a particular case of the mod $p$ Lück approximation for abstract finitely generated subgroups of free-by-$\mathbb{Z}_p$ pro-$p$ groups. |
| title | Sylvester domains and pro-$p$ groups |
| topic | Group Theory Rings and Algebras 20E18, 20E22, 16S35, 16S85 |
| url | https://arxiv.org/abs/2402.14130 |