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We show that there are contact 3-manifolds of support genus one which admitinfinitely many Stein fillings, but do not admit arbitrarily large ones. TheseStein fillings arise from genus-1 allowable Lefschetz fibrations with distincthomology groups, all filling a fixed minimal genus open book supporting theboundary contact 3-manifold. In contrast, we observe that there are onlyfinitely many possibilities for the homology groups of Stein fillings of agiven contact 3-manifold with support genus zero. We also show that there are4-strand braids which admit infinitely many distinct Hurwitz classes ofquasipositive factorizations, yielding in particular an infinite family ofknotted complex analytic annuli in the 4-ball bounding the same transverse linkup to transverse isotopy. These realize the smallest possible examples in termsof the number of boundary components a genus-1 mapping class and the number ofstrands a braid can have with infinitely many positive/quasipositivefactorizations.

Fillings of genus-1 open books and 4-braids