Groups of order 64 and non-homeomorphic double Kodaira fibrations with the same biregular invariants
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| Format: | Preprint |
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2024
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| author | Polizzi, Francesco Sabatino, Pietro |
| author_facet | Polizzi, Francesco Sabatino, Pietro |
| contents | Let $Σ_b$ be a closed Riemann surface of genus $b$. We investigate finite quotients $G$ of the pure braid group on two strands $\mathsf{P}_2(Σ_b)$ which do not factor through $π_1(Σ_b \times Σ_b)$. Building on our previous work on some special systems of generators on finite groups that we called \emph{diagonal double Kodaira structures}, we prove that, if $G$ has not order $32$, then $|G| \geq 64$, and we completely classify the cases where equality holds. In the last section, as a geometric application of our algebraic results, we construct two $3$-dimensional families of double Kodaira fibrations having the same biregular invariants and the same Betti numbers but different fundamental group. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_08260 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Groups of order 64 and non-homeomorphic double Kodaira fibrations with the same biregular invariants Polizzi, Francesco Sabatino, Pietro Algebraic Geometry Group Theory Geometric Topology 14J29, 14J25, 20D15 Let $Σ_b$ be a closed Riemann surface of genus $b$. We investigate finite quotients $G$ of the pure braid group on two strands $\mathsf{P}_2(Σ_b)$ which do not factor through $π_1(Σ_b \times Σ_b)$. Building on our previous work on some special systems of generators on finite groups that we called \emph{diagonal double Kodaira structures}, we prove that, if $G$ has not order $32$, then $|G| \geq 64$, and we completely classify the cases where equality holds. In the last section, as a geometric application of our algebraic results, we construct two $3$-dimensional families of double Kodaira fibrations having the same biregular invariants and the same Betti numbers but different fundamental group. |
| title | Groups of order 64 and non-homeomorphic double Kodaira fibrations with the same biregular invariants |
| topic | Algebraic Geometry Group Theory Geometric Topology 14J29, 14J25, 20D15 |
| url | https://arxiv.org/abs/2412.08260 |