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 fibered 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 $\text{Spin}^c$-structure $\mathfrak{s}$ (i.e., one that is in a `corner' of the support of $SFH$) via a nice and taut sutured manifold decomposition even when $H_2(M) \neq 0$, assuming the corresponding group $SFH(M,\gamma,\mathfrak{s})$ has non-trivial Euler characteristic.