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. Theorem 1.2. Let M be an oriented closed connected graph manifold (in the sense of Definition 2.7) with residually finite fundamental group. Then M = M ∞ Z = 0. More generally, let (Γ j ) j∈N be a descending chain of finite index subgroups of π 1 (M ) with trivial intersection and let (M j ) j∈N be the corresponding tower of finite coverings. Then M = lim j→∞ M j Z [π 1 (M ) : Γ j ] = 0.

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We provide new computations in bounded cohomology: A group is boundedly acyclic if its bounded cohomology with trivial real coefficients is zero in all positive degrees. We show that there exists a continuum of finitely generated non-amenable boundedly acyclic groups and that there exists a finitely presented boundedly acyclic group that is universal in the sense that it contains all finitely presented groups.On the other hand, we construct a continuum of finitely generated groups, whose bounded cohomology has uncountable dimension in all degrees greater than or equal to 2. Countable non-amenable groups with these two extreme properties were previously known to exist, but these constitute the first finitely generated examples.Finally, we show that various algorithmic problems on bounded cohomology are undecidable.

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