Formality properties of finitely generated groups and Lie algebras

TL;DR

This paper investigates the formality properties of finitely generated groups by analyzing associated Lie algebras, using models and expansions, with applications to various topological and group-theoretic examples.

Contribution

It provides a comprehensive study of how different notions of formality behave under various algebraic and topological operations, linking group models to Lie algebra structures.

Findings

Formality properties are preserved under certain algebraic operations.

The 1-minimal model effectively relates group properties to Lie algebra structures.

Examples include torsion-free nilpotent groups and fundamental groups of manifolds.

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.