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

TL;DR

This paper proves an inequality relating volume and L2-Betti numbers of aspherical manifolds, introduces new vanishing theorems, and establishes a gap theorem linking minimal volume to L2-Betti number vanishing.

Contribution

It provides a proof of a volume versus L2-Betti number inequality using measured equivalence relations and extends vanishing theorems for L2-Betti numbers of aspherical manifolds.

Findings

Established an inequality between volume and L2-Betti numbers.

Proved new vanishing theorems for L2-Betti numbers.

Derived a gap theorem linking minimal volume to L2-Betti number vanishing.

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.