Infinitesimal finiteness obstructions

Papadima, Ştefan; Suciu, Alexander I.

doi:10.1112/jlms.12169

Cited by 5 publications

(20 citation statements)

References 40 publications

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“…Theorem 7.2 admits the following generalization: if G is a finitely generated group, and if (A, d) is a connected dga with dim A 1 < ∞ whose 1-minimal model is isomorphic to M (G; k), then the Malcev Lie algebra m(G; k) is isomorphic to the completion with respect to the degree filtration of the Lie algebra h(A, d) := lie((A 1 ) * )/ im((d 1 ) * + µ * A ) . Proofs of this result are given in [2,63]; related results can be found in [4,5,66].…”

Section: Filtered-formality and 1-formalitymentioning

confidence: 89%

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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: Filtered-formality and 1-formalitymentioning

confidence: 89%

“…Given a finitely presented group G, the solvable quotients G/G (i) need not be finitely presented (see [63]). Thus, finding presentations for the Chen Lie algebra gr(G/G (i) ) can be an arduous task.…”

Section: 5mentioning

confidence: 99%

Formality properties of finitely generated groups and Lie algebras

Suciu

Wang

2019

Forum Mathematicum

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“…A related result (based on the above proof) is given in Proposition 4.9 from [34]. In fact, the same proof as in Corollary 5.3 works for all universal solvable quotients: Corollary 5.4.…”

Section: Jump Loci and Alexander Invariantsmentioning

confidence: 76%

Residually finite rationally p groups

Koberda

Suciu

2019

Commun. Contemp. Math.

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In this article we develop the theory of residually finite rationally p (RFRp) groups, where p is a prime. We first prove a series of results about the structure of finitely generated RFRp groups (either for a single prime p, or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFRp groups, which we use to study the boundary manifolds of algebraic curves CP 2 and in C 2 . We show that boundary manifolds of a large class of curves in C 2 (which includes all line arrangements) have RFRp fundamental groups, whereas boundary manifolds of curves in CP 2 may fail to do so.

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“…For q " 1, the aforementioned properties of the space X can be interpreted purely in terms of the Malcev Lie algebra of it fundamental group, mpπq. Indeed, as shown in [40], X admits a 1-finite 1-model if and only if mpπq is the lower central series completion of a finitely presented Lie algebra L. More stringently, as shown in the foundational work of Quillen [41] and Sullivan [49], X is 1-formal if and only if L can be chosen to be a quadratic Lie algebra.…”

Section: Now Letmentioning

confidence: 99%

“…The existence of such a model puts rather stringent constraints on the group π. One such constraint is given in [40,Theorem 1.5].…”

Section: Theorem 43 ([25]mentioning

confidence: 99%

Cohomology jump loci of 3-manifolds

Suciu

2019

Preprint

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The cohomology jump loci of a space X are of two basic types: the characteristic varieties, defined in terms of homology with coefficients in rank one local systems, and the resonance varieties, constructed from information encoded in either the cohomology ring, or an algebraic model for X. We explore here the geometry of these varieties and the delicate interplay between them in the context of closed, orientable 3-dimensional manifolds and link complements. The classical multivariable Alexander polynomial plays an important role in this analysis. As an application, we derive some consequences regarding the formality and the existence of finite-dimensional models for such 3-manifolds.

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Infinitesimal finiteness obstructions

Cited by 5 publications

References 40 publications

Formality properties of finitely generated groups and Lie algebras

Formality properties of finitely generated groups and Lie algebras

Residually finite rationally p groups

Cohomology jump loci of 3-manifolds

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