The simplicial volume of 3‐manifolds with boundary

We provide sharp lower bounds for the simplicial volume of compact 3‐manifolds in terms of the simplicial volume of their boundaries. As an application, we compute the simplicial volume of several classes of 3‐manifolds, including handlebodies and products of surfaces with the interval. Our results provide the first exact computation of the simplicial volume of a compact manifold whose boundary has positive simplicial volume. We also compute the minimal number of tetrahedra in a (loose) triangulation of the product of a surface with the interval.