A note on the fundamental group of Kodaira fibrations

The fundamental group $\pi$ of a Kodaira fibration is, by definition, the extension of a surface group $\Pi_b$ by another surface group $\Pi_g$, i.e. $$1 \rightarrow \Pi_g \rightarrow \pi \rightarrow \Pi_b \rightarrow 1.$$ Conversely, we can inquire about what conditions need to be satisfied by a group of that sort in order to be the fundamental group of a Kodaira fibration. In this short note we collect some restriction on the image of the classifying map $m \colon \Pi_b \to \Gamma_g$ in terms of the coinvariant homology of $\Pi_g$. In particular, we observe that if $\pi$ is the fundamental group of a Kodaira fibration with relative irregularity $g-s$, then $g \leq 1+ 6s$, and we show that this effectively constrains the possible choices for $\pi$, namely that there are group extensions as above that fail to satisfy this bound, hence cannot be the fundamental group of a Kodaira fibration. In particular this provides examples of symplectic $4$--manifolds that fail to admit a K\"ahler structure for reasons that eschew the usual obstructions.