“ L 2 - Betti numbers of mapping tori and groups ” by

We prove the following two conjectures of Gromov. Firstly, all L2-Betti numbers of a manifold fibered over S1 are trivial. Secondly, the first L2-Betti number of a finitely presented group Γ vanishes provided that Γ is an extension {1} −→ ∆ −→ Γ −→ π −→ {1} of finitely presented groups such that ∆ is infinite and π contains Z as a subgroup. We conclude for such a group Γ that its deficiency is less than or equal to one and that any closed 4-manifold with Γ as fundamental group satisfies χ(M) ≥ |σ(M)|.