We study a filtered vector space A d (n) over a field k of characteristic 0, which consists of Jacobi diagrams of degree d on n oriented arcs for each n, d ≥ 0. We consider an action of the automorphism group Aut(Fn) of the free group Fn of rank n on the space A d (n), which is induced by an action of handlebody groups on bottom tangles. The action of Aut(Fn) on A d (n) induces an action of the general linear group GL(n, k) on the associated graded vector space of A d (n), which is regarded as the vector space B d (n) consisting of open Jacobi diagrams. Moreover, the Aut(Fn)-action on A d (n) induces an action on B d (n) of the associated graded Lie algebra gr(IA(n)) of the IA-automorphism group IA(n) of Fn with respect to its lower central series. We use an irreducible decomposition of B d (n) and computation of the gr(IA(n))-action on B d (n) to study the Aut(Fn)-module structure of A d (n). In particular, we consider the case where d = 2 in detail and give an indecomposable decomposition of A 2 (n). We also consider a functor A d , which includes the Aut(Fn)-module structure of A d (n) for all n.Contents 30 References 31

We consider an action of the automorphism group Aut(Fn) of the free group Fn of rank n on the filtered vector space A d (n) of Jacobi diagrams of degree d on n oriented arcs. This action induces on the associated graded vector space of A d (n), which is identified with the space B d (n) of open Jacobi diagrams, an action of the general linear group GL(n, Z) and an action of the graded Lie algebra of the IA-automorphism group of Fn associated with its lower central series. We use these actions on B d (n) to study the Aut(Fn)-module structure of A d (n). In particular, we consider the case where d = 2 in detail and give an indecomposable decomposition of A 2 (n). We also construct a polynomial functor A d of degree 2d from the opposite category of the category of finitely generated free groups to the category of filtered vector spaces, which includes the Aut(Fn)-module structure of A d (n) for all n 0.Résumé. -Nous considérons une action du groupe d'automorphisme s Aut(Fn) du groupe libre Fn de rang n sur l'espace vectoriel filtré A d (n) des diagrammes de Jacobi de degré d sur n arcs orientés. Cette action induit sur l'espace vectoriel gradué associé de A d (n), qui est identifié à l'espace B d (n) des diagrammes de Jacobi ouverts, une action du groupe linéaire général GL(n, Z) et une action de l'algèbre de Lie graduée du groupe d'automorphismes IA de Fn associée à sa série centrale inférieure. Nous utilisons ces actions sur B d (n) pour étudier la structure de Aut(Fn)-module de A d (n). En particulier, nous considérons en détail le cas où d = 2 et donnons une décomposition indécomposable de A 2 (n). Nous construisons également un foncteur polynomial A d de degré 2d de la catégorie opposée de la catégorie des groupes libres finiment engendrés à la catégorie des espaces vectoriels filtrés, qui inclut la structure de Aut(Fn)-module de A d (n) pour tout n 0.

The automorphism group Aut(Fn) of the free group Fn acts on a space A d (n) of Jacobi diagrams of degree d on n oriented arcs. We study the Aut(Fn)-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 GL(n, Z) and an action of the graded Lie algebra gr(IA(n)) of the IA-automorphism group IA(n) of Fn associated with its lower central series. We extend the action of gr(IA(n)) to an action of the associated graded Lie algebra of the Andreadakis filtration of the endomorphism monoid of Fn. By using this action, we study the Aut(Fn)-module structure of A d (n). We obtain a direct decomposition of A d (n) as Aut(Fn)-modules for general d, which is indecomposable for d = 3, 4. Moreover, we obtain the radical of A d (n) for general d and the socle of A 3 (n). Contents 1. Introduction 1 2. Preliminaries 6 3. Andreadakis filtration E * (n) of End(F n ) 11 4. Action of gr(E * (n)) on B d (n) 17 5. Contraction map 26 6. Correspondence between the map βr d,k and the map γ r d,k 30 7. The GL(V n )-module structure of B d (n) 33 8. The Aut(F n )-module structure of A d (n) 40 9. The functor A d 50 Appendix A. Presentation of the category A L 51 References 54

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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