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Amenable covers, volume and L2-Betti numbers of aspherical manifolds
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Roman Sauer
Published/Copyright:
October 12, 2009
Abstract
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.
Received: 2007-07-04
Published Online: 2009-10-12
Published in Print: 2009-November
© Walter de Gruyter Berlin · New York 2009
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Articles in the same Issue
- Transition maps at non-resonant hyperbolic singularities are o-minimal
- Amenable covers, volume and L2-Betti numbers of aspherical manifolds
- ℒ-optimal transportation for Ricci flow
- Deformation theory of representations of prop(erad)s II
- The Bass and topological stable ranks of and
- A derived approach to geometric McKay correspondence in dimension three
