2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/163806We provide a proof for an inequality between volume and L2-Betti numbers of aspherical manifolds for which Gromov outlined a strategy based on general ideas of Connes. The implementation of that strategy involves measured equivalence relations, Gaboriau's theory of L2-Betti numbers of R-simplicial complexes, and other themes of measurable group theory. Further, we prove new vanishing theorems for L2-Betti numbers that generalize a classical result of Cheeger and Gromov. As one of the corollaries, we obtain a gap theorem which implies vanishing of L2-Betti numbers of an aspherical manifold when its minimal volume is sufficiently small.minor corrections; to appear in J. Reine Angew. Math (Crelle)Algebraic TopologyDifferential GeometryDynamical Systems22D20; 53C20; 58J22text