Length functions on mapping class groups and simplicial volumes of mapping tori

Let M be a closed orientable manifold. We introduce two numerical invariants, called filling volumes, on the mapping class group MCG(M) of M, which are defined in terms of filling norms on the space of singular boundaries on M, both with real and with integral coefficients. We show that filling volumes are length functions on MCG(M), we prove that the real filling volume of a mapping class f is equal to the simplicial volume of the corresponding mapping torus E_f, while the integral filling volume of f is not smaller than the stable integral simplicial volume of E_f. . We discuss several vanishing and non-vanishing results for the filling volumes. As applications, we show that the hyperbolic volume of 3-dimensional mapping tori is not subadditive with respect to their monodromy, and that the real and the integral filling norms on integral boundaries are often non-biLipschitz equivalent.