Resonance varieties, Chen ranks and formality properties of finitely generated groups

by He Wang

Institution: Northeastern University
Year: 2016
Keywords: Alexander invariants; Chen ranks; formality; pure virtual braid groups; pure welded braid groups; resonance varieties
Posted: 02/05/2017
Record ID: 2104537
Full text PDF: http://hdl.handle.net/2047/D20213217


Formality is a topological property that arises from the rational homotopy theory developed by Quillen and Sullivan in 70's. Roughly speaking, the rational homotopy type of a formal space is determined by its cohomology algebra. In this thesis, we explore the graded-formality, filtered-formality, and 1-formality of finitely-generated groups, by studying various Lie algebras over a field of characteristic 0 attached to such groups, including the associated graded Lie algebra, the holonomy Lie algebra, and the Malcev Lie algebra. We explain how these notions behave with respect to split injections, coproducts, direct products, and how they are inherited by solvable and nilpotent quotients.; 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.