Surface braid groups, Finite Heisenberg covers and double Kodaira fibrations

We exhibit new examples of double Kodaira fibrations by using finite Galois covers a product $\Sigma_b \times \Sigma_b$, where $\Sigma_b$ is smooth projective curve genus $b \geq 2$. Each cover obtained providing an explicit group epimorphism from the pure braid $\mathsf{P}_2(\Sigma_b)$ to some Heisenberg group. In this way, we are able show that every $b$ base fibration; moreover, number pairwise non-isomorphic fibred surfaces fibering over fixed at least $\boldsymbol{\omega}(b+1)$, $\boldsymbol{\omega} \colon \mathbb{N} \to \mathbb{N}$ stands for arithmetic function counting distinct prime factors positive integer. As particular case our general construction, obtain real $4$-manifold signature $144$ can be realized as surface bundle $2$, with fibre $325$, in two different ways. This provides (to knowledge) first "double" solution problem Kirby's list low-dimensional topology.