Integral filling volume, complexity, and integral simplicial volume of $3$-dimensional mapping tori

We show that the integral filling volume of a Dehn twist f on a closed oriented surface vanishes, i.e., that the integral simplicial volume of the mapping torus with monodromy f^{n} grows sublinearly with respect to n . We deduce a complete characterization of mapping classes on surfaces with vanishing integral filling volume and, building on results by Purcell and Lackenby on the complexity of mapping tori, we show that, in dimension three, complexity and integral simplicial volume are not Lipschitz equivalent.