Abstract: A Kähler group is a group which is the fundamental group of a compact Kähler manifold. A key tool in the study of Kähler groups are homomorphisms to fundamental groups of closed hyperbolic surfaces (surface groups). Given a finite collection of such homomorphisms, one can study their image in the direct product of the surface groups. I will explain a classification of the possible images for products of three surface groups, answering a question of Delzant and Gromov for this case. This provides constraints on Kähler groups. I will then discuss how these constraints can be applied to gain insights into the nature of Kodaira fibrations which admit more than two distinct fibrations. This is joint work with Pierre Py.