2014
DOI: 10.4171/jems/447
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The abelianization of the Johnson kernel

Abstract: Abstract. We prove that the first complex homology of the Johnson subgroup of the Torelli group T g is a non-trivial, unipotent T g -module for all g ≥ 4 and give an explicit presentation of it as a Sym • H 1 (T g , C)-module when g ≥ 6. We do this by proving that, for a finitely generated group G satisfying an assumption close to formality, the triviality of the restricted characteristic variety implies that the first homology of its Johnson kernel K is a nilpotent module over the corresponding Laurent polyno… Show more

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Cited by 17 publications
(24 citation statements)
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References 41 publications
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“…As in the introduction, K g,n denotes the subgroup of T g,n generated by Dehn twists on bounding simple closed curves (BSCCs). 8 The following result is a special case of a result [26, Thm. A] of Putman.…”
Section: Variation On a Theme Of Putmanmentioning
confidence: 90%
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“…As in the introduction, K g,n denotes the subgroup of T g,n generated by Dehn twists on bounding simple closed curves (BSCCs). 8 The following result is a special case of a result [26, Thm. A] of Putman.…”
Section: Variation On a Theme Of Putmanmentioning
confidence: 90%
“…We can thus compute the map (8) log t A : J/J 2 → J 3 /J 4 by computing t A (γ) − γ for a set of loops γ that span H. First, if γ is the image of a loop in (S ′ , x), then t A (γ) = γ, which implies that the restriction of (8) to H ′ is trivial. To compute the restriction of (8) to H ′′ , we compute t A on loops γ = ρµρ −1 that follow a fixed path ρ in S ′ from x to a point x ′ ∈ A and then follow a loop µ in S ′′ and return to x along ρ −1 .…”
Section: Commuting Elementsmentioning
confidence: 99%
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“…In topology, it is known that the rank 1 jump loci V i 1 pM q control delicate finiteness properties of Alexander-type invariants of M . This fact was used in [4,3] to obtain significant results on certain important subgroups of mapping class groups of closed Riemann surfaces.…”
Section: Introduction and Statement Of Resultsmentioning
confidence: 99%
“…It is readily seen that this definition agrees with the one from [14], modulo a degree shift by 2. Work from [13,14,16] as well as [2,Proposition 2.4], implies that…”
Section: Infinitesimal Alexander Invariantmentioning
confidence: 99%