Intersection subgroups of complex hyperplane arrangements

Abstract

Let A be a central arrangement of hyperplanes in C n , let M( A) be the complement of A , and let L( A) be the intersection lattice of A . For X in L( A) we set A X={H∈ A: H⫆X} , and A/X={H/X: H∈ A X} , and A X={H∩X: H∈ A\\ A X} . We exhibit natural embeddings of M( A X) in M( A) that give rise to monomorphisms from π 1(M( A X)) to π 1(M( A)) . We call the images of these monomorphisms intersection subgroups of type X and prove that they form a conjugacy class of subgroups of π 1(M( A)) . Recall that X in L( A) is modular if X+Y is an element of L( A) for all Y in L( A) . We call X in L( A) supersolvable if there exists a chain 0⫅X 1⫅⋯⫅X d=X in L( A) such that X μ is modular and dim X μ=μ for all μ=1,…,d . Assume that X is supersolvable and view π 1(M( A X)) as an intersection subgroup of type X of π 1(M( A)) . Recall that the commensurator of a subgroup S in a group G is the set of a in G such that S∩aSa −1 has finite index in both S and aSa −1 . The main result of this paper is the characterization of the centralizer, the normalizer, and the commensurator of π 1(M( A X)) in π 1(M( A)) . More precisely, we exhibit an embedding of π 1(M( A X)) in π 1(M( A)) and prove: (1) π 1(M( A X))∩π 1(M( A X))={1} and π 1(M( A X)) is included in the centralizer of π 1(M( A X)) in π 1(M( A)) ; (2) the normalizer is equal to the commensurator and is equal to the direct product of π 1(M( A X)) and π 1(M( A X)) ; (3) the centralizer is equal to the direct product of π 1(M( A X)) and the center of π 1(M( A X)) . Our study starts with an investigation of the projection p :M( A)→M( A/X) induced by the projection C n→ C n/X . We prove in particular that this projection is a locally trivial C ∞ fibration if X is modular, and deduce some exact sequences involving the fundamental groups of the complements of A , of A/X , and of some (affine) arrangement A z 0 X .