Alexander invariants of complex hyperplane arrangements

Abstract

Let A \mathcal {A} be an arrangement of n n complex hyperplanes. The fundamental group of the complement of A \mathcal {A} is determined by a braid monodromy homomorphism, α : F s → P n \alpha :F_{s}\to P_{n} . Using the Gassner representation of the pure braid group, we find an explicit presentation for the Alexander invariant of A \mathcal {A} . From this presentation, we obtain combinatorial lower bounds for the ranks of the Chen groups of A \mathcal {A} . We also provide a combinatorial criterion for when these lower bounds are attained.