  Two standard invariants used to study the fundamental group G of the
complement X of a hyperplane arrangement are the Malcev completion of G and the
cohomology groups of X with coefficients in rank one local systems. In this
paper, we develop a tool that unifies these two approaches. This tool is the
Malcev completion S_p of G relative to a homomorphism p from G into (C^*)^N.
This is a prosolvable group that is tightly controlled by the cohomology groups
of X with coefficients in rank one local systems. The prounipotent radical U_p
of the relative completion S_p corresponds to a pronilpotent Lie algebra u_p.
We provide an example of a hyperplane complement X for which this algebra is
not quadratically presented. In addition, we show that if X is a hyperplane
complement and Y is a subtorus of the character torus, then S_p is
combinatorially determined for general p in Y. Finally, we show that the
relative completion S_p is generally constant over subvarieties of the
character torus.
