Formality properties of finitely generated groups and Lie algebras

Abstract

We explore the graded and filtered formality properties of finitely generated groups by studying the various Lie algebras over a field of characteristic 0 attached to such groups, including the Malcev Lie algebra, the associated graded Lie algebra, the holonomy Lie algebra, and the Chen Lie algebra. We explain how these notions behave with respect to split injections, coproducts, direct products, as well as field extensions, and how they are inherited by solvable and nilpotent quotients. A key tool in this analysis is the 1-minimal model of the group, and the way this model relates to the aforementioned Lie algebras. Another approach to formality is provided by Taylor expansions from the group to the completion of the associated graded algebra of the group ring. We illustrate our approach with examples drawn from a variety of group-theoretic and topological contexts, such as finitely generated torsion-free nilpotent groups, link groups, and fundamental groups of Seifert fibered manifolds.

41 pages; accepted for publication in Forum Mathematicum

References in corpus (8)

• The pure braid groups and their relatives
• Pure virtual braids, resonance, and formality
• Cup products, lower central series, and holonomy Lie algebras
• Quadratic-linear duality and rational homotopy theory of chordal arrangements
• The topology of compact Lie group actions through the lens of finite models
• On the geometry and topology of partial configuration spaces of Riemann surfaces
• Infinitesimal finiteness obstructions
• Cohomology jump loci of 3-manifolds

Cited by in corpus (18)

• Koszul algebras and quadratic duals in Galois cohomology
• Local systems on complements of arrangements of smooth, complex algebraic hypersurfaces
• The pure braid groups and their relatives
• Around the tangent cone theorem
• Pure virtual braids, resonance, and formality
• Cup products, lower central series, and holonomy Lie algebras
• Chen ranks and resonance varieties of the upper McCool groups
• On the universal ellipsitomic KZB connection
• The topology of compact Lie group actions through the lens of finite models
• Reduced resonance schemes and Chen ranks
• Homology, lower central series, and hyperplane arrangements
• Cohomology jump loci of 3-manifolds
• Formality and finiteness in rational homotopy theory
• Taylor expansions of groups and filtered-formality
• Milnor fibrations of arrangements with trivial algebraic monodromy
• Effective computation of degree bounded minimal models for GCDA's
• Alexander invariants and cohomology jump loci in group extensions
• Koszulity, supersolvability, and Stirling representations