Discriminantal bundles, arrangement groups, and subdirect products of free groups
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Discriminantal bundles, arrangement groups, and subdirect products of free groups Daniel C. Cohen1
· Michael J. Falk2 · Richard C. Randell3
In memory of Bill Arvola and S¸ tefan Papadima Received: 17 February 2019 / Revised: 24 November 2019 / Accepted: 4 May 2020 © Springer Nature Switzerland AG 2020
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 , B , D , 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 representationsof the arrangement group π1 (M). We describe the kernel of the product ρX : 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 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.
This project was begun during the program on hyperplane arrangements at Mathematical Sciences Research Institute in 2004, and was continued during conferences at the Pacific Institute of Mathematical Sciences, the Fields Institute, the University of Zaragoza, and Hokkaido University, and during the program on configuration spaces at Centro di Ricerca Matematica Ennio de Giorgi in Pisa. The authors thank these institutions and the meeting organizers for their support and hospitality. We also thank several anonymous referees for their careful reading and helpful suggestions. Extended author information available on the last page of the article
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Keywords Arrangement · Discriminantal, decomposable, pure braid group · Brunnian braid · Subdirect product · Cohomological finiteness type Mathematics Subject Classification 20F36 · 32S22 · 52C35 · 55R80
1 Introduction Let A = {H1 , . . . , Hn } be an arrangement of affine hyperplanes in C, with comn plement M = M(A) = C − i=1 Hi . Suppose F = { f 1 , . . . , f μ } is a set of complex-val
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