TY - JOUR

T1 - Discriminantal bundles, arrangement groups, and subdirect products of free groups

AU - Cohen, Daniel C.

AU - Falk, Michael J.

AU - Randell, Richard C.

N1 - Funding Information:
We are grateful to Graham Denham, Daniel Matei, and Alex Suciu for many helpful discussions. Theorem?4.6 and Proposition?4.12 were formulated with the help of Daniel Malcolm when he was an undergraduate research student of the second author.

PY - 2020/9/1

Y1 - 2020/9/1

N2 - We construct bundles Ek(A, F) → M over the complement M of a complex hyperplane arrangement A, depending on an integer k⩾ 1 and a set F= { f1, … , fμ} of continuous functions fi: 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 Aℓ, Bℓ, Dℓ, F4, 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 π1(M). We describe the kernel of the product ρX: G→ ∏ S∈XGS of homomorphisms of a finitely-generated group G onto quotient groups GS 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→ GS split in a compatible way, we show the image of ρX 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 π1(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 π1(M) has a conjugation-free presentation or it is not residually nilpotent.

AB - We construct bundles Ek(A, F) → M over the complement M of a complex hyperplane arrangement A, depending on an integer k⩾ 1 and a set F= { f1, … , fμ} of continuous functions fi: 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 Aℓ, Bℓ, Dℓ, F4, 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 π1(M). We describe the kernel of the product ρX: G→ ∏ S∈XGS of homomorphisms of a finitely-generated group G onto quotient groups GS 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→ GS split in a compatible way, we show the image of ρX 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 π1(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 π1(M) has a conjugation-free presentation or it is not residually nilpotent.

KW - Arrangement

KW - Brunnian braid

KW - Cohomological finiteness type

KW - Discriminantal, decomposable, pure braid group

KW - Subdirect product

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U2 - 10.1007/s40879-020-00412-1

DO - 10.1007/s40879-020-00412-1

M3 - Article

AN - SCOPUS:85086703176

VL - 6

SP - 751

EP - 789

JO - European Journal of Mathematics

JF - European Journal of Mathematics

SN - 2199-675X

IS - 3

ER -