How large is the braid monodromy of low-genus Lefschetz fibrations?

Abstract

Given a genus $g$ smooth Lefschetz fibration $π: M \to S^2$ with singular locus $Δ\subseteq S^2$, we describe the subgroup $\operatorname{Br}(π)$ of the spherical braid group $\operatorname{Mod}(S^2,Δ)$ consisting of braids admitting a lift to a fiber-preserving diffeomorphism of $M$. We develop general methods for showing that the index $[\operatorname{Mod}(S^2,Δ) : \operatorname{Br}(π)]$ is infinite. As an application of our methods, we prove that $[\operatorname{Mod}(S^2,Δ) : \operatorname{Br}(π)] = \infty$ when $g = 1$, when $π$ is expressible as a self-fiber sum when $g \geq 2$, or when $π$ is a holomorphic genus $g = 2$ Lefschetz fibration whose vanishing cycles are nonseparating. In the genus $g = 1$ case, we relate the subgroup $\operatorname{Br}(π)$ to the action of $\operatorname{Mod}(S^2,Δ)$ on the $\operatorname{SL}_2$-character variety for $S^2 \setminus Δ$ and provide an alternate proof of the first application via recent work of Lam--Landesman--Litt.