2000/10/04 by Daniel Matei, Alexander I. Suciu
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Finite Group Theory Research #math.CO #math.GR #math.GT #msc:20E07 #msc:20J05 #msc:52C35 #msc:57M05
paper · pdf · doi:10.1155/s107379280210907x
published as International Math. Research Notices 2002:9 (2002), 465-503 · 34 pages, 3 figures
arxiv created 2000/10/04 · openalex publication_date 2002/01/01 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a finitely-generated group G, and a finite group Γ, Philip Hall defined δΓ(G) to be the number of factor groups of G that are isomorphic to Γ. We show how to compute the Hall invariants by cohomological and combinatorial methods, when G is finitely-presented, and Γ belongs to a certain class of metabelian groups. Key to this approach is the stratification of the character variety, Hom(G,K*), 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 δΓ(G). In the process, we compute the distribution of prime-index, normal subgroups K ◃ G according to dimKH1(K;K), provided char K ≠ |G : K|. 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 ℂ2, or transverse planes in ℝ4.