Taylor expansions of groups and filtered-formality

Abstract

Let G be a finitely generated group, and let $$\\Bbbk {G}$$ 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 $$\\Bbbk {G}$$ 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.