Discriminantal bundles, arrangement groups, and subdirect products of free groups

Abstract

We construct bundles $$E_k({{\mathscr {A}}},{{\mathscr {F}}}) \rightarrow M$$ over the complement M of a complex hyperplane arrangement $${{\mathscr {A}}}$$ , depending on an integer $$k \geqslant 1$$ and a set $${{\mathscr {F}}}=\{f_1, \ldots , f_\mu \}$$ of continuous functions $$f_i :M \rightarrow {{\mathbb {C}}}$$ whose differences are nonzero on M, generalizing the configuration space bundles arising in the Lawrence–Krammer–Bigelow representation of the pure braid group. We display such families $${{\mathscr {F}}}$$ for rank two arrangements, reflection arrangements of types $$A_\ell $$ , $$B_\ell $$ , $$D_\ell $$ , $$F_4$$ , and for arrangements supporting multinet structures with three classes, with the resulting bundles having nontrivial monodromy around each hyperplane. The construction extends to arbitrary arrangements by pulling back these bundles along products of inclusions arising from subarrangements of these types. We then consider the faithfulness of the resulting representations of the arrangement group $$\pi _1(M)$$ . We describe the kernel of the product $$\rho _{{\mathscr {X}}}:G \rightarrow \prod _{S \in {{\mathscr {X}}}} G_S$$ of homomorphisms of a finitely-generated group G onto quotient groups $$G_S$$ determined by a family $${{\mathscr {X}}}$$ of subsets of a fixed set of generators of G, extending a result of Theodore Stanford about Brunnian braids. When the projections $$G \rightarrow G_S$$ split in a compatible way, we show the image of $$\rho _{{\mathscr {X}}}$$ is normal with free abelian quotient, and identify the cohomological finiteness type of G. These results apply to some well-studied arrangements, implying several qualitative and residual properties of $$\pi _1(M)$$ , including an alternate proof of a result of Artal, Cogolludo, and Matei on arrangement groups and Bestvina–Brady groups, and a dichotomy for a decomposable arrangement $${{\mathscr {A}}}$$ : either $$\pi _1(M)$$ has a conjugation-free presentation or it is not residually nilpotent.