Finite quotients of surface braid groups and double Kodaira fibrations

TL;DR

This paper explores finite quotients of surface braid groups, called pure braid quotients, and demonstrates their application in constructing new double Kodaira fibrations, advancing understanding in algebraic geometry and topology.

Contribution

It provides new insights into pure braid quotients of surface braid groups and introduces novel methods for constructing double Kodaira fibrations.

Findings

Identification of specific non-abelian finite groups as pure braid quotients

New constructions of double Kodaira fibrations using these groups

Extension of previous results on surface braid group quotients

Abstract

Let $\Sigma_b$ be a closed Riemann surface of genus $b$. We give an account of some results obtained in the recent papers \cite{CaPol19, Pol20, PolSab21} and concerning what we call here \emph{pure braid quotients},namely non-abelian finite groups appearing as quotients of the pure braid group on two strands $\mathsf{P}_2(\Sigma_b)$. We also explain how these groups can be used in order to provide new constructions of double Kodaira fibrations.