Amenable covers, volume and L2-Betti numbers of aspherical manifolds

Details Amenable covers, volume and L2-Betti numbers of aspherical manifolds Amenable covers, volume and L2-Betti numbers of aspherical manifolds

2006-05-23

arxiv.org/abs/math/0605627v3

Mathematics

Algebraic Topology

minor corrections; to appear in J. Reine Angew. Math (Crelle)

Scientific paper

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.

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