Filtrations and cohomology on graph products
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910004875886592 |
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| author | Hamza, Oussama |
| author_facet | Hamza, Oussama |
| contents | Let $p$ be a prime. We resolve a question posed by Mináč-Rogelstad-Tân. We relate the Zassenhaus and the lower central series of pro-$p$ groups under a torsion-freeness condition.
We also study graph products of (pro-$p$) groups under natural assumptions. In particular, we compute their graded Lie algebras associated with the previous filtrations, as well as their cohomology over $\mathbb{F}_p$. Our approach relies on various filtrations of amalgamated products, as studied in Leoni's PhD thesis. Explicit examples are provided using the Koszul property.
As a concrete application, we compute the cohomology over $\mathbb{F}_p$ and the graded Lie algebras associated with the filtrations of graph products of fundamental groups of surfaces. These groups furnish new examples satisfying the torsion-freeness condition, which arises in the question of Mináč-Rogelstad-Tân. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21784 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Filtrations and cohomology on graph products Hamza, Oussama Group Theory 20F05, 20F14, 20F40, 16S10, 17A45 Let $p$ be a prime. We resolve a question posed by Mináč-Rogelstad-Tân. We relate the Zassenhaus and the lower central series of pro-$p$ groups under a torsion-freeness condition. We also study graph products of (pro-$p$) groups under natural assumptions. In particular, we compute their graded Lie algebras associated with the previous filtrations, as well as their cohomology over $\mathbb{F}_p$. Our approach relies on various filtrations of amalgamated products, as studied in Leoni's PhD thesis. Explicit examples are provided using the Koszul property. As a concrete application, we compute the cohomology over $\mathbb{F}_p$ and the graded Lie algebras associated with the filtrations of graph products of fundamental groups of surfaces. These groups furnish new examples satisfying the torsion-freeness condition, which arises in the question of Mináč-Rogelstad-Tân. |
| title | Filtrations and cohomology on graph products |
| topic | Group Theory 20F05, 20F14, 20F40, 16S10, 17A45 |
| url | https://arxiv.org/abs/2601.21784 |