Spaces with vanishing $l\sp 2$-homology and their fundamental groups (after Farber and Weinberger)

TL;DR

This paper explores manifolds with fundamental groups whose universal covers have vanishing L2-homology, providing counterexamples to the zero in the spectrum conjecture and extending the understanding of L2-cohomology in high-dimensional manifolds.

Contribution

It establishes the existence of high-dimensional manifolds with specified fundamental groups whose universal covers have zero L2-cohomology in all degrees, strengthening previous counterexamples.

Findings

Counterexamples to the zero in the spectrum conjecture for certain groups.

Construction of high-dimensional manifolds with vanishing L2-cohomology.

Extension of Farber and Weinberger's ideas to new manifold classes.

Abstract

The "zero in the spectrum conjecture" asserted (in its strongest form) that for any manifold M zero should be in the l2-spectrum of the Laplacian (on forms) of the universal covering of M, i.e. that at least one (unreduced) L2-cohomology group of (the universal covering of) M is non-zero. Farber and Weinberger gave the first counterexamples to this conjecture. In this paper, using their fundamental idea to show the following stronger version of this result: Let G be a finitely presented group and suppose that the homology groups H_k(G,\ell^2(G)) are zero for k=0,1,2. For every dimension n\ge 6 there is a closed manifold M of dimension n and with fundamental group G such that the L2-cohomology of (the universal covering of) M vanishes in all degrees.