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ACTIONS OF AUTOMORPHISM GROUPS OF FREE GROUPS ON SPACES OF JACOBI DIAGRAMS. II

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  09 June 2022

Mai Katada Open the ORCID record for Mai Katada [Opens in a new window]
Mai Katada*
Affiliation:
Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan
*
katada.mai.36s@st.kyoto-u.ac.jp
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Abstract

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 $A_3(n)$.

Keywords

Jacobi diagramsautomorphism groups of free groupsgeneral linear groupsIA-automorphism groups of free groupsAndreadakis filtration

MSC classification

Primary: 20F12: Commutator calculus
Secondary: 20F28: Automorphism groups of groups
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 23 , Issue 1 , January 2024 , pp. 1 - 69
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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