  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.
