6533b860fe1ef96bd12c3c34

RESEARCH PRODUCT

Intersection subgroups of complex hyperplane arrangements

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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 .