Hall invariants, homology of subgroups, and characteristic varieties

Abstract

Given a finitely-generated group G, and a finite group Γ, Philip Hall defined δ_Γto 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 Γbelongs 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 δ_Γ(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.

34 pages, 3 figures

References in corpus (4)

• Geometry of cohomology support loci for local systems I
• Characteristic varieties of arrangements
• The Braid Monodromy of Plane Algebraic Curves and Hyperplane Arrangements
• Homotopy types of complements of 2-arrangements in R^4

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• Formality, Alexander invariants, and a question of Serre
• Fundamental groups, Alexander invariants, and cohomology jumping loci
• Torsion in Milnor fiber homology
• The line geometry of resonance varieties
• Counting homomorphisms onto finite solvable groups
• Fundamental groups of real arrangements and torsion in the lower central series quotients
• Cohomology jump loci of 3-manifolds
• Multivariable Alexander invariants of hypersurface complements
• Milnor fibrations of arrangements with trivial algebraic monodromy
• Computing untwisted Dijkgraaf-Witten invariants for arborescent links
• Resonance varieties over fields of positive characteristic
• On the number of harmonic frames
• Complex hyperplane arrangements