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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 bounded by 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.

Fillings of Genus–1 Open Books and 4–Braids