  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.
