Taylor expansions of groups and filtered-formality

Abstract

Let $G$ be a finitely generated group, and let $\mathbb{k}G$ be its group algebra over a field of characteristic $0$. A Taylor expansion is a certain type of map from $G$ to the degree completion of the associated graded algebra of $\mathbb{k}G$ which generalizes the Magnus expansion of a free group. The group $G$ is said to be filtered-formal if its Malcev Lie algebra is isomorphic to the degree completion of its associated graded Lie algebra. We show that $G$ is filtered-formal if and only if it admits a Taylor expansion, and derive some consequences.

arXiv comment19 pages; accepted for publication in the European Journal of Mathematics

References in corpus (12)

• Topology and geometry of cohomology jump loci
• Chen Lie algebras
• Basis-conjugating automorphisms of a free group and associated Lie algebras
• Finite Type Invariants of w-Knotted Objects I: w-Knots and the Alexander Polynomial
• Geometric and algebraic aspects of 1-formality
• Formality properties of finitely generated groups and Lie algebras
• Homotopy types of complements of 2-arrangements in R^4
• Chen ranks and resonance
• The pure braid groups and their relatives
• The Pure Virtual Braid Group Is Quadratic
• Cup products, lower central series, and holonomy Lie algebras
• Chen ranks and resonance varieties of the upper McCool groups