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

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

2010-08-02

arxiv.org/abs/1008.0417v2

Mathematics

Geometric Topology

29 pages, 9 figures

Scientific paper

The Lawrence-Krammer-Bigelow representation of the braid group arises from the monodromy representation on the twisted homology of the fiber of a certain fiber bundle in which the base and total space are complements of braid arrangements, and the fiber is the complement of a discriminantal arrangement. We present a more general version of this construction and use it to construct nontrivial bundles on the complement of an arbitrary arrangement \A\ whose fibers are products of discriminantal arrangements. This leads us to consider the natural homomorphism $\rho_\X$ from the arrangement group $G(\A)=\pi_1(\C^\ell - \bigcup \A)$ to the product of groups $G(\A_X), X \in \X,$ corresponding to a set \X\ of rank-two flats. Generalizing an argument of T. Stanford, we describe the kernel in terms of iterated commutators, when generators of $G(\A_X), X \in \X,$ can be chosen compatibly. We use this to derive a test for injectivity of $\rho_\X.$ We show $\rho_\X$ is injective for several well-studied decomposable arrangements. If \A\ is central, the homomorphism $\rho_\X$ induces a natural homomorphism $\bar{\rho}_\X$ from the projectivized group $\bar{G}(\A)$ into the product $\prod_{X\in \X} \bar{G}(\A_X),$ whose factors are free groups. We show $\bar{\rho}_\X$ is injective if and only if $\rho_\X$ is. In this case $\bar{G}$ is isomorphic to a specific finitely-presented, combinatorially-determined subdirect product of free groups. In particular $\bar{G}$ is residually free, residually torsionfree nilpotent, a-T-menable, and linear. We show the image of $\bar{\rho}_\X$ is a normal subgroup with free abelian quotient, and compute the rank of the quotient in terms of the incidence graph of \X\ with \A. When $\rho_\X$ is injective, we conclude $\bar{G}$ is of type $F_{m-1}$ and not of type $F_m,$ $m=|\X|.$

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