Resonance varieties, Chen ranks and formality properties of finitely generated groups
Permanent URL:
http://hdl.handle.net/2047/D20213217
Suciu, Alexandru I. (Committee member)
Martsinkovsky, Alexander (Committee member)
Porter, Richard (Committee member)
Igusa, Kiyoshi (Committee member)
We investigate the varied relationships among several algebraic and geometric invariants of finitely-generated groups, including the aforementioned Lie algebras, commutative differential graded algebras, Chen Lie algebras, Alexander-type invariants as well as resonance varieties and characteristic varieties. Significant results arise from the study of the interactions between theses objects, e.g., the tangent cone theorem of Dimca, Papadima and Suciu, and the Chen ranks formula conjectured by Suciu and proved by Cohen and Schenck.
For a finitely-presented group, we give an explicit formula for the cup product in low degrees, and an algorithm for computing the holonomy Lie algebra, using a Magnus expansion method. We also give a presentation for the Chen Lie algebra of a filtered-formal group, and discuss various approaches to computing the ranks of the graded objects under consideration.
We apply our techniques to several families of braid-like groups: the pure braid groups, the pure welded braid groups, the virtual pure braid groups, as well as their 'upper' variants. We also discuss several natural homomorphisms between these groups, and various ways to distinguish among them. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as 1-relator groups, finitely generated torsion-free nilpotent groups, link groups, and fundamental groups of Seifert fibered manifolds.
Chen ranks
formality
pure virtual braid groups
pure welded braid groups
resonance varieties
Finite groups
Representations of groups
Resonance -- Mathematics
Homotopy theory
Braid theory
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