Taylor expansions of groups and filtered-formality
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Taylor expansions of groups and filtered-formality Alexander I. Suciu1
· He Wang1
To the memory of S¸ tefan Papadima, 1953–2018 Received: 31 May 2019 / Accepted: 3 November 2019 © Springer Nature Switzerland AG 2019
Abstract Let G be a finitely generated group, and let kG 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 kG 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. Keywords Taylor expansion · Hopf algebra · Chen iterated integrals · Malcev Lie algebra · Filtered-formality · 1-formality · Residually torsion-free nilpotent group · Automorphism groups of free groups Mathematics Subject Classification 20F40 · 16T05 · 16W70 · 17B70 · 20F14 · 20J05 · 55P62
1 Introduction 1.1 Expansions of groups Group expansions were first introduced by Magnus in [30], in order to show that finitely generated free groups are residually nilpotent. This technique has been generalized and
Alexander I. Suciu is supported in part by the Simons Foundation collaboration Grant for mathematicians 354156.
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He Wang [email protected]; [email protected] Alexander I. Suciu [email protected] https://web.northeastern.edu/suciu/
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Department of Mathematics, Northeastern University, Boston, MA 02115, USA
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A.I. Suciu, H. Wang
used in many ways. For instance, the exponential expansion of a free group was used to give a presentation for the Malcev Lie algebra of a finitely presented group by Papadima [39] and Massuyeau [34]. Expansions of pure braid groups and their applications in knot theory have been studied since the 1980s by several authors, see for instance Kohno’s papers [24–26]. Lin studied in [29] expansions of fundamental groups of smooth manifolds, using Chen’s theory [12] of formal power series connections and their induced monodromy representations. More generally, Bar-Natan explored in [4] the Taylor expansions of an arbitrary ring. Let G be a finitely generated group, and fix a coefficient field k of characteristic zero. We let gr(kG) be the associated graded algebra of kG with respect to the filtration by powers of the augmentation ideal, and we let gr(kG) be the degree completion of this algebra. Developing an idea from [4], we say that a map E : G → gr(kG) is a multiplicative expansion of G if the induced algebra morphism, E : kG → gr(kG), is filtration-preserving and induces the identity at the associated graded level. Such a map E is called a Taylor expansion if it sends each element of G to a group-like element of the Hopf algebra gr(kG). 1.2 Expansions and filtered-formality Once again, let G be a finitely generated group. The concept of filtered-formality relates an object from rational homotopy theory to a group-theoret
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