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

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.

💡 Research Summary

The paper revisits the “zero in the spectrum conjecture,” which in its strongest form predicts that for any closed manifold M the Laplacian on the universal cover (\widetilde M) has 0 in its spectrum, equivalently that some unreduced (L^2)-cohomology group of (\widetilde M) is non‑zero. Farber and Weinberger produced the first counterexamples, but their construction relied on very special fundamental groups.
The authors extend the Farber–Weinberger idea to any finitely presented group (G) satisfying a stringent ℓ²‑homological vanishing condition: \