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

Surface braid groups, finite Heisenberg covers and double Kodaira fibrations