The spectrum of simplicial volume

New constructions in group homology allow us to manufacture high-dimensional manifolds with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of simplicial volumes of orientable closed connected manifolds is dense in  $$\mathbb {R}_{\ge 0}$$ . In dimension 4 we prove that every non-negative rational number is the simplicial volume of some orientable closed connected 4-manifold. Our group theoretic results relate stable commutator length to the $$l^1$$ -semi-norm of certain singular homology classes in degree 2. The output of these results is translated into manifold constructions using cross-products and Thom realisation.