Mathematics – Group Theory
Scientific paper
2000-10-04
International Math. Research Notices 2002:9 (2002), 465-503
Mathematics
Group Theory
34 pages, 3 figures
Scientific paper
10.1155/S107379280210907X
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.
Matei Daniel
Suciu Alexander I.
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