The simplicial volume is a homotopy invariant of manifolds; this talk is about the simplicial volume of a Davis' manifolds, obtained from the so-called reflection group trick, which is a powerful method for constructing aspherical manifolds. I will describe an approach based on the study of triangulations of spheres and simplicial maps between them. This approach also presents connections with the theory of graph minors. No knowledge about simplicial volume or Davis' reflection group trick is expected from the audience.