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arXiv · 2412.08260

Groups of order 64 and non-homeomorphic double Kodaira fibrations with the same biregular invariants

Francesco Polizzi·Pietro Sabatino
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Abstract

Let $\Sigma_b$ be a closed Riemann surface of genus $b$. We investigate finite quotients $G$ of the pure braid group on two strands $\mathsf{P}_2(\Sigma_b)$ which do not factor through $\pi_1(\Sigma_b \times \Sigma_b)$. 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.

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Francesco Polizzi, Pietro Sabatino. 2024-12-11. Groups of order 64 and non-homeomorphic double Kodaira fibrations with the same biregular invariants. https://doi.org/10.1007/s10231-026-01691-3

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