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Taylor expansions of groups and filtered-formality. (English) Zbl 07270595

Summary: 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.

MSC:

20F40 Associated Lie structures for groups
16T05 Hopf algebras and their applications
16W70 Filtered associative rings; filtrational and graded techniques
17B70 Graded Lie (super)algebras
20F14 Derived series, central series, and generalizations for groups
20J05 Homological methods in group theory
55P62 Rational homotopy theory
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