On the low-dimensional cohomology groups of the IA-automorphism group of the free group of rank three

Satoh, Takao

doi:10.1017/s0013091521000171

Cited by 2 publications

(2 citation statements)

References 37 publications

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“…Day-Putman [10] obtained a finite set of generators of H 2 (IA n , Z) as GL(n, Z)-representations. For n = 3, Satoh [28] detected a nontrivial irreducible subrepresentation of H 2 (IA 3 , Q), which can not be detected by the Johnson homomorphism. However, the second homology of IA n has not been completely determined.…”

Section: Introductionmentioning

confidence: 98%

Stable rational homology of the IA-automorphism groups of free groups

Katada¹

2022

Preprint

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The rational homology of the IA-automorphism group IAn of the free group Fn is still mysterious. We study the quotient of the rational homology of IAn that is obtained as the image of the map induced by the abelianization map, which we call the Albanese homology of IAn. We obtain a representation-stable GL(n, Q)-subquotient of the Albanese homology of IAn, which conjecturally coincides with the entire Albanese homology of IAn. In particular, we obtain a lower bound of the dimension of the Albanese homology of IAn for each homological degree in a stable range. Moreover, we determine the entire third Albanese homology of IAn for n ≥ 9. We also study the Albanese homology of an analogue of IAn to the outer automorphism group of Fn. Contents Introduction Preliminaries Abelian cycles in HAlbanese cohomology of IA n 9. Albanese homology of IO n 10. The third Albanese homology of IO n 11. The third Albanese homology of IA n 12. Problems and Perspectives Appendix A. Properties of Albanese homology and cohomology References

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Section: Introductionmentioning

confidence: 98%

Stable rational homology of the IA-automorphism groups of free groups

Katada¹

2022

Preprint

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“…Pettet [25] determined the second Albanese homology of IA n . Satoh [28] obtained a subquotient GL(3, Z)-representation of H 2 (IA 3 , Q) which is not contained in the Albanese homology of IA 3 . We determined the Albanese homology in degree 3 and studied it in higher degrees [15].…”

Section: Introductionmentioning

confidence: 99%

Actions of Automorphism Groups of Free Groups on Spaces of Jacobi Diagrams. Ii

Katada

2022

J. Inst. Math. Jussieu

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The automorphism group $\operatorname {Aut}(F_n)$ of the free group $F_n$ acts on a space $A_d(n)$ of Jacobi diagrams of degree d on n oriented arcs. We study the $\operatorname {Aut}(F_n)$ -module structure of $A_d(n)$ by using two actions on the associated graded vector space of $A_d(n)$ : an action of the general linear group $\operatorname {GL}(n,\mathbb {Z})$ and an action of the graded Lie algebra $\mathrm {gr}(\operatorname {IA}(n))$ of the IA-automorphism group $\operatorname {IA}(n)$ of $F_n$ associated with its lower central series. We extend the action of $\mathrm {gr}(\operatorname {IA}(n))$ to an action of the associated graded Lie algebra of the Andreadakis filtration of the endomorphism monoid of $F_n$ . By using this action, we study the $\operatorname {Aut}(F_n)$ -module structure of $A_d(n)$ . We obtain an indecomposable decomposition of $A_d(n)$ as $\operatorname {Aut}(F_n)$ -modules for $n\geq 2d$ . Moreover, we obtain the radical filtration of $A_d(n)$ for $n\geq 2d$ and the socle of $A_3(n)$ .

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On the low-dimensional cohomology groups of the IA-automorphism group of the free group of rank three

Cited by 2 publications

References 37 publications

Stable rational homology of the IA-automorphism groups of free groups

Stable rational homology of the IA-automorphism groups of free groups

Actions of Automorphism Groups of Free Groups on Spaces of Jacobi Diagrams. Ii

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