Sutured Floer homology, fibrations, and taut depth one foliations

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 fibred classes in H-1(M). Furthermore, we show that the SFH of a balanced sutured manifold (M, gamma) detects which classes in H-1(M) admit a taut depth one foliation such that the only compact leaves are the components of R(gamma). The latter had been proved earlier by the first author under the extra assumption that H-2(M) = 0. The main technical result is that we can obtain an extremal Spin(c)-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 H-2(M) not equal 0, assuming the corresponding group SFH(M, gamma, s) has non-trivial Euler characteristic