Twisted homological stability for extensions and automorphism groups of free nilpotent groups

Szymik, Markus

doi:10.1017/is014005031jkt267

Cited by 6 publications

(14 citation statements)

References 31 publications

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“…The inspiration to use the above bootstrapping method came from a paper of Szymik [25] where this method is applied to the closely related automorphism groups of free nilpotent groups. Szymik considers the extensions…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…With this result in place, we can present a proof of Theorem B. We use a classical spectral sequence argument which we learnt from [25,Theorem 1•9]. THEOREM B.…”

Section: Homological Stability For the Groups G1 (K)mentioning

confidence: 99%

“…For every k 1, the FI-module ker ρ(k) is finitely generated and takes values in free abelian groups.Proof. There is an isomorphism (cf [25,. Proposition 3•1]) ker ρ(k) ∼ = K (k) ∼ = (Z • ) * ⊗ gr k F ∼ = gr 1 F ⊗ gr k F, where (Z • ) * isthe dual FI-module, obtained by composing Z • : FI op → Ab with the functor Hom(−, Z) : Ab op → Ab.…”

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confidence: 99%

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On the quotients of mapping class groups of surfaces by the Johnson subgroups

Zeman

2019

Math. Proc. Camb. Phil. Soc.

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We study quotients of mapping class groups Γ g,1 of oriented surfaces with one boundary component by terms of their Johnson filtrations, and we show that the homology of these quotients with suitable systems of twisted coefficients stabilises as the genus of the surface goes to infinity. We also compute the stable (co)homology with constant rational coefficients for one family of such quotients.Remark 2.3. C -Mod is abelian, with kernels and cokernels computed pointwise. Hence it makes good sense to talk of sub-C -modules, quotient C -modules etc. Now suppose further that the category C is strict monoidal, where the monoidal product ⊕ is given by n ⊕ m = n + m on objects, and that the monoidal unit 0 is initial in C .Definition 2.4. We will call such C a category with objects the naturals, or CON for short.Consider the functor R := 1 ⊕ (−) : C → C . Since 0 is initial, the unique arrow 0 → 1 induces a natural transformation ρ : Id ⇒ R from the identity Id = 0 ⊕ (−) to R.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

“…With this result in place, we can present a proof of Theorem B. We use a classical spectral sequence argument which we learnt from [25,Theorem 1•9]. THEOREM B.…”

Section: Homological Stability For the Groups G1 (K)mentioning

confidence: 99%

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confidence: 99%

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On the quotients of mapping class groups of surfaces by the Johnson subgroups

Zeman

2019

Math. Proc. Camb. Phil. Soc.

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“…Remark 6.2. The existence of a stable range for the homology of the automorphism groups (with polynomial coefficients) has been established in [Szy14,Thm. 4.5], even integrally.…”

Section: Outer Automorphism Groupsmentioning

confidence: 99%

The rational stable homology of mapping class groups of universal nil-manifolds

Szymik¹

2019

Annales De l'Institut Fourier

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We compute the rational stable homology of the automorphism groups of free nilpotent groups. These groups interpolate between the general linear groups over the ring of integers and the automorphism groups of free groups, and we employ functor homology to reduce to the abelian case. As an application, we also compute the rational stable homology of the outer automorphism groups and of the mapping class groups of the associated aspherical nil-manifolds in the TOP, PL, and DIFF categories. MSC: 20J05, 20E36 (19M05, 18A25, 18G40)

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“…between that and (1.2), based on earlier work [Szy14,Szy19] on the theories Nil c of nilpotent groups of a given class c, letting c → ∞, in the spirit of 'nilpotent mathematics' (see [Sha91]).…”

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confidence: 99%

Generalizations of Loday's assembly maps for Lawvere's algebraic theories

Bohmann¹,

Szymik²

2021

Preprint

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Loday's assembly maps approximate the K-theory of group rings by the K-theory of the coefficient ring and the corresponding homology of the group. We present a generalization that places both ingredients on the same footing. Building on Elmendorf-Mandell's multiplicativity results and our earlier work, we show that the K-theory of Lawvere theories is lax monoidal. This result makes it possible to present our theory in a user-friendly way without using higher categorical language. It also allows us to extend the idea to new contexts and set up a non-abelian interpolation scheme, raising novel questions. Numerous examples illustrate the scope of our extension.

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Twisted homological stability for extensions and automorphism groups of free nilpotent groups

Cited by 6 publications

References 31 publications

On the quotients of mapping class groups of surfaces by the Johnson subgroups

On the quotients of mapping class groups of surfaces by the Johnson subgroups

The rational stable homology of mapping class groups of universal nil-manifolds

Generalizations of Loday's assembly maps for Lawvere's algebraic theories

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