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