Intersection subgroups of complex hyperplane arrangements
نویسنده
چکیده
A X g. We exhibit natural embeddings of M(A X) in M(A) that give rise to monomorphisms from 1 (M(A X)) to 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 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 is modular and dimX = for all = 1; : : : ; d. Assume that X is supersolvable and view 1 (M(A X)) as an intersection subgroup of type X of 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 nite 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 1 (M(A X)) in 1 (M(A)). More precisely, we exhibit an embedding of 1 (M(A X)) in 1 (M(A)) and prove: 1) 1 (M(A X)) \ 1 (M(A X)) = f1g and 1 (M(A X)) is included in the centralizer of 1 (M(A X)) in 1 (M(A)); 2) the normalizer is equal to the commensurator and is equal to the direct product of 1 (M(A X)) and 1 (M(A X)); 3) the centralizer is equal to the direct product of 1 (M(A X)) and the center of 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 1 bration if X is modular, and deduce some exact sequences involving the fundamental groups of the complements of A, of A=X, and of some (aane) arrangement A X z 0 .
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