# Taylor expansions of groups and filtered-formality

## TL;DR

This paper explores the concept of Taylor expansions for finitely generated groups and establishes their equivalence with the property of filtered-formality, providing new insights into the algebraic structure of such groups.

## Contribution

It introduces the notion of Taylor expansions for groups and proves that a group is filtered-formal if and only if it admits such an expansion.

## Key findings

- Filtered-formality is characterized by the existence of a Taylor expansion.
- Taylor expansions generalize the Magnus expansion for free groups.
- The paper derives consequences of the equivalence between filtered-formality and Taylor expansions.

## Abstract

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