Integral approximation of simplicial volume of graph manifolds

Graph manifolds are manifolds that decompose along tori into pieces with a tame $S^1$-structure. In this paper, we prove that the simplicial volume of graph manifolds (which is known to be zero) can be approximated by integral simplicial volumes of their finite coverings. This gives a uniform proof of the vanishing of rank gradients, Betti number gradients and torsion homology gradients for graph manifolds.