A spectral sequence to compute L2-Betti numbers of groups and groupoids

TL;DR

This paper introduces a spectral sequence for computing L2-Betti numbers of groups and groupoids, proving the Hopf-Singer conjecture in specific cases and deriving new vanishing theorems and computations.

Contribution

It constructs a novel spectral sequence for L2-cohomology of groupoids and applies it to prove the Hopf-Singer conjecture for certain aspherical manifolds, also deriving new results.

Findings

Proves the Hopf-Singer conjecture for aspherical manifolds with poly-surface fundamental groups.

Provides new vanishing theorems for L2-Betti numbers.

Offers explicit computations and obstructions related to measured groupoids.

Abstract

We construct a spectral sequence for L2-type cohomology groups of discrete measured groupoids. Based on the spectral sequence, we prove the Hopf-Singer conjecture for aspherical manifolds with poly-surface fundamental groups. More generally, we obtain a permanence result for the Hopf-Singer conjecture under taking fiber bundles whose base space is an aspherical manifold with poly-surface fundamental group. As further sample applications of the spectral sequence, we obtain new vanishing theorems and explicit computations of L2-Betti numbers of groups and manifolds and obstructions to the existence of normal subrelations in measured equivalence relations.