Cohomology Jumping Loci and Relative Malcev Completion

Two standard invariants used to study the fundamental group of the complement X of a hyperplane arrangement are the Malcev (i.e. unipotent) completion of its fundamental group π1(X,x0) and the cohomology groups H (X,Lρ) 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 of π1(X,x0) relative to a homomorphism ρ : π1(X,x0) → (C ) . This is a prosolvable group that generalizes the Malcev completion of π1(X,x0) and is tightly controlled by the cohomology groups H(X,L ρ k1 1 ···ρ kN N ).