Sutured Floer homology, fibrations, and taut depth one foliations

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For an oriented irreducible 3-manifold M with non-empty toroidal boundary, we describe how sutured Floer homology (SFH) can be used to determine all fibered classes in H¹(M). Furthermore, we show that the SFH of a balanced sutured manifold (M,γ) detects which classes in H¹(M) admit a taut depth one foliation such that the only compact leaves are the components of R(γ). The latter had been proved earlier by the first author under the extra assumption that H₂(M)=0. The main technical result is that we can obtain an extremal Spinᶜ-structure s (i.e., one that is in a `corner' of the support of SFH) via a nice and taut sutured manifold decomposition even when H₂(M) ≠ 0, assuming the corresponding group SFH(M,γ,s) has non-trivial Euler characteristic.