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Autori: D. Matei, A.I. Suciu
Editorial: International Mathematics Research Notices, 2002 (9), p.465-503, 2002.
Given a finitely-generated group G, and a finite group Gamma, Philip Hall defined delta_Gamma to be the number of factor groups of G that are isomorphic to Gamma. We show how to compute the Hall invariants by cohomological and combinatorial methods, when G is finitely-presented, and Gamma belongs to a certain class of metabelian groups. Key to this approach is the stratification of the character variety by the jumping loci of the cohomology of G, with coefficients in rank 1 local systems over a suitably chosen field K. Counting relevant torsion points on these “characteristic” subvarieties gives delta_Gamma(G). In the process, we compute the distribution of prime-index, normal subgroups K of G according to the dimension of the the first homology group of K with K coefficients, provided char K does not divide the index of K in G. In turn, we use this distribution to count low-index subgroups of G. We illustrate these techniques in the case when G is the fundamental group of the complement of an arrangement of either affine lines in C^2, or transverse planes in R^4.
Cuvinte cheie: Hall invariants, metabelian groups, characteristic varieties, cohomology of groups, low-index subgroups, fundamental groups