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

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 $|G|\ge 64$, 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.

The paper consists of 24 pages. There is also an appendix (8 pages) containing the tables of finite groups. arXiv admin note: text overlap with arXiv:2102.04963