  We 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.
