Infinitesimal presentations of the Torelli groups

Hain, Richard

doi:10.1090/s0894-0347-97-00235-x

Cited by 177 publications

(293 citation statements)

References 41 publications

(51 reference statements)

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“…The following definition will play a key role in the sequel. This notion appears in the work of Bezrukavnikov [4] and Hain [31], as well as in the work of Calaque et. al [7] under the name of 'formality', and in the work of Lee [45], under the name of 'weak-formality'.…”

Section: 3mentioning

confidence: 72%

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Formality properties of finitely generated groups and Lie algebras

Suciu

Wang

2019

Forum Mathematicum

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We explore the graded-formality 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. 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.

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Section: 3mentioning

confidence: 72%

“…As shown by in Hain in [31,32] the Torelli groups in genus 4 or higher are 1-formal, but the Torelli group in genus 3 is filtered-formal, yet not graded-formal. The next two examples show that there are groups which are graded-formal but not filtered-formal.…”

Section: Massey Productsmentioning

confidence: 89%

Formality properties of finitely generated groups and Lie algebras

Suciu

Wang

2019

Forum Mathematicum

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“…As noted by Hain [43] in the case of the Torelli group of a Riemann surface and proved by Berceanu and Papadima [15] in full generality, a stronger assumption leads to a stronger conclusion.…”

Section: Residual Propertiesmentioning

confidence: 81%

“…A lot is also known about the residual properties of the pure braid-like groups, especially as they relate to the lower central series. For instance, a theorem of Berceanu and Papadima [15], which uses work of Andreadakis [1] and an idea of Hain [43], shows that the groups IA n are residually torsion-free nilpotent, for all n. Table 1 Hilbert series, Koszulness, and formality of pure braid-like groups.…”

Section: Lie Algebras Lcs Ranks and Formalitymentioning

confidence: 99%

The Pure Braid Groups and Their Relatives

Suciu

Wang

2017

Perspectives in Lie Theory

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In this mostly survey paper, we investigate the resonance varieties, the lower central series ranks, and the Chen ranks, as well as the residual and formality properties of several families of braid-like groups: the pure braid groups P n , the welded pure braid groups wP n , the virtual pure braid groups vP n , as well as their 'upper' variants, wP + n and vP + n . We also discuss several natural homomorphisms between these groups, and various ways to distinguish among the pure braid groups and their relatives.

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“…There is a broad range of remarkable results for the Johnson homomorphisms of a mapping class group. (For example, see [7] and [15], [16], [17]. )…”

Section: Johnson Filtrationmentioning

confidence: 99%

The Johnson filtration of the McCool stabilizer subgroup of the automorphism group of a free group

Satoh

2011

Proc. Amer. Math. Soc.

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Abstract. Let F n be a free group of rank n with basis x 1 , x 2 , . . . , x n . We denote by S n the subgroup of the automorphism group of F n consisting of automorphisms which fix each of x 2 , . . . , x n and call it the McCool stabilizer subgroup. Let IS n be a subgroup of S n consisting of automorphisms which induce the identity on the abelianization of F n . In this paper, we determine the group structure of the lower central series of IS n and its graded quotients. Then we show that the Johnson filtration of S n coincides with the lower central series of IS n .

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Infinitesimal presentations of the Torelli groups

Cited by 177 publications

References 41 publications

Formality properties of finitely generated groups and Lie algebras

Formality properties of finitely generated groups and Lie algebras

The Pure Braid Groups and Their Relatives

The Johnson filtration of the McCool stabilizer subgroup of the automorphism group of a free group

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