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

Daniel C. Cohen, Michael J. Falk, Richard C. Randell

Published 2010-08-02, updated 2019-04-16Version 3

A generating set for a complex hyperplane arrangement A is a collection of continuous functions on the complement M of A whose differences are non-vanishing on M. A generating set F gives rise to a space of extensions E_k(A,F) of A which is the total space of a bundle over M. One can use such bundles to construct representations of the fundamental group of M analogous to the Lawrence-Krammer-Bigelow representation of the pure braid group. We construct generating sets for several families of arrangements. In this context we analyze the kernel of the homomorphism p_X from an arbitrary group G with finite generating set Y to the product of quotients G_S of G corresponding to an antichain X of 2^Y. Generalizing an argument of T. Stanford concerning Brunnian braids, we show the kernel of p_X is generated by iterated commutators that are transverse to X in a certain sense, provided the projections of G to G_S split compatibly. Under this hypothesis, we derive a test for the injectivity of p_X that depends only on 2- and 3-fold commutators of generators. We show that p_X is injective for some well-studied decomposable arrangements. If A is central and X consists of rank two flats, the homomorphism p_X induces a homomorphism from the quotient of G by Z to a product of free groups. We show this map is injective if and only if p_X is, provided the incidence graph L of X with A is connected. The image is a normal subgroup, and the cokernel is equal to H^1(L,Z). Using results of Meier-Meinert-Van Wyk on subdirect products of free groups, we then identify the cohomological finiteness type of the image in terms of L, and deduce the cohomological finiteness type of several arrangement groups. Finally we show that a decomposable arrangement group either has a conjugation-free presentation or is not residually nilpotent.