  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.
