### Abstract

We construct bundles E_{k}(A, F) → M over the complement M of a complex hyperplane arrangement A, depending on an integer k⩾ 1 and a set F= { f_{1}, … , f_{μ}} of continuous functions f_{i}: M→ 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 F for rank two arrangements, reflection arrangements of types A_{ℓ}, B_{ℓ}, D_{ℓ}, 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 π_{1}(M). We describe the kernel of the product ρ_{X}: G→ ∏ _{S}_{∈}_{X}G_{S} of homomorphisms of a finitely-generated group G onto quotient groups G_{S} determined by a family X of subsets of a fixed set of generators of G, extending a result of Theodore Stanford about Brunnian braids. When the projections G→ G_{S} split in a compatible way, we show the image of ρ_{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 π_{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 A: either π_{1}(M) has a conjugation-free presentation or it is not residually nilpotent.

Original language | English (US) |
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Journal | European Journal of Mathematics |

DOIs | |

State | Accepted/In press - 2020 |

### Keywords

- Arrangement
- Brunnian braid
- Cohomological finiteness type
- Discriminantal, decomposable, pure braid group
- Subdirect product

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

*European Journal of Mathematics*. https://doi.org/10.1007/s40879-020-00412-1