Extra-special quotients of surface braid groups and double Kodaira fibrations with small signature

TL;DR

This paper investigates special finite group quotients of surface braid groups using diagonal double Kodaira structures, establishing minimal group sizes and constructing new examples of double Kodaira fibrations with small signature.

Contribution

It introduces the concept of diagonal double Kodaira structures, characterizes extra-special groups as minimal cases, and constructs new families of double Kodaira fibrations with small signature.

Findings

Finite groups with diagonal double Kodaira structures have order at least 32.

Equality in size occurs if and only if the group is extra-special.

Constructed two 3-dimensional families of double Kodaira fibrations with signature 16.

Abstract

We study some special systems of generators on finite groups, introduced in previous work by the first author and called "diagonal double Kodaira structures", in order to investigate non-abelian, finite quotients of the pure braid group on two strands $\mathsf{P}_2(\Sigma_b)$, where $\Sigma_b$ is a closed Riemann surface of genus $b$. In particular, we prove that, if a finite group $G$ admits a diagonal double Kodaira structure, then $|G|\geq 32$, and equality holds if and only if $G$ is extra-special. In the last section, as a geometrical application of our algebraic results, we construct two $3$-dimensional families of double Kodaira fibrations having signature $16$. Such surfaces are different from the ones recently constructed by Lee, L\"onne and Rollenske and, as far as we know, they provide the first examples of positive-dimensional families of double Kodaira fibrations with small signature.