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

TL;DR

The paper classifies certain finite quotients of the pure braid group related to double Kodaira fibrations, and constructs examples with identical invariants but different fundamental groups.

Contribution

It provides a classification of finite quotients of the pure braid group with specific properties and constructs new double Kodaira fibrations with identical invariants but distinct fundamental groups.

Findings

Finite quotients of the pure braid group have order at least 64 unless they are of order 32.

Complete classification of cases where the order of these quotients equals 64.

Constructed families of double Kodaira fibrations with same invariants but different fundamental groups.

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.