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} 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|.