THE $\Sigma^3$-CONJECTURE FOR METABELIAN GROUPS

Harlander, Jens; Kochloukova, Dessislava H.

doi:10.1112/s0024610702004088

Cited by 4 publications

(4 citation statements)

References 19 publications

(30 reference statements)

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“…Even though it is generally quite difficult to describe Σ k (G) and Σ k (G; Z), there are some important classes of groups for which the Sigma invariants can be determined, for example right-angled Artin groups, see [21], and Thompson's group F , see [4]. For more information and applications of these invariants see, for example, [2,3,5,6,19].…”

Section: Bieri-neumann-strebel-renz Invariantsmentioning

confidence: 99%

Closed 1-forms in topology and geometric group theory

Farber¹,

Geoghegan²,

Schütz³

2010

Russ. Math. Surv.

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In this article we describe relations of the topology of closed 1-forms to the group theoretic invariants of Bieri-Neumann-Strebel-Renz. Starting with a survey, we extend these Sigma invariants to finite CWcomplexes and show that many properties of the group theoretic version have analogous statements. In particular we show the relation between Sigma invariants and finiteness properties of certain infinite covering spaces. We also discuss applications of these invariants to the Lusternik-Schnirelmann category of a closed 1-form and to the existence of a nonsingular closed 1-form in a given cohomology class on a high-dimensional closed manifold.To S.P. Novikov on the occasion of his 70-th birthday

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Section: Bieri-neumann-strebel-renz Invariantsmentioning

confidence: 99%

Closed 1-forms in topology and geometric group theory

Farber¹,

Geoghegan²,

Schütz³

2010

Russ. Math. Surv.

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“…[18]). Более подробную информацию об этих инвариантах и их приложениях см., например, в [8], [9], [19]- [21].…”

Section: инварианты бьери-неймана-штребеля-ренцаunclassified

Замкнутые 1-Формы В Топологии И Геометрической Теории Групп

Farber¹,

Farber²,

Гейган³

et al. 2010

Uspekhi Mat. Nauk

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“…In general the homological invariant Σ m (G, Z) is an open subset of the unit sphere S(G) and Σ m (G, Z) determines which subgroups of G above the commutator are of homological type F P m [6]. The homological and homotopical Σ m -invariants of a group are quite difficult to calculate but they are known for right-angled Artin groups [20], the R. Thompson group F [7], metabelian groups of finite Prüfer rank [19] and for split extensions metabelian groups if m = 3 [16].…”

Section: Introductionmentioning

confidence: 99%

Homological Finiteness Properties of Wreath Products

Bartholdi

Cornulier

Kochloukova

2015

The Quarterly Journal of Mathematics

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Abstract. We study the homological finiteness properties F Pm of wreath products Γ = H ≀ X G. We show that, when H has infinite abelianization, Γ has type F Pm if and only if both G and H have type F Pm and G acts (diagonally) on X i with stabilizers of type F P m−i and with finitely many orbits for all 1 ≤ i ≤ m.If furthermore H is torsion-free we give a criterion for Γ to be Bredon-F Pm with respect to the class of finite subgroups of Γ.Finally, when H has infinite abelianization and χ : Γ → R is a non-zero homomorphism with χ(H) = 0, we classify when [χ] belongs to the BieriNeumann-Strebel-Renz invariant Σ m (Γ, Z).

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THE $\Sigma^3$-CONJECTURE FOR METABELIAN GROUPS

Abstract: The Σ 3 -conjecture for metabelian groups is proved in the split extension case.

Cited by 4 publications

References 19 publications

Closed 1-forms in topology and geometric group theory

Closed 1-forms in topology and geometric group theory

Замкнутые 1-Формы В Топологии И Геометрической Теории Групп

Homological Finiteness Properties of Wreath Products

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