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On the structures of the Johnson cokernels of the basis-conjugating automorphism groups of free groups

Part of: Special aspects of infinite or finite groups Connections with homological algebra and category theory Representation theory of groups

Published online by Cambridge University Press:  21 July 2025

Naoya Enomoto  and
Takao Satoh
Naoya Enomoto
Affiliation:
The University of Electro-Communications, Tokyo, 182-8585, Japan
Takao Satoh*
Affiliation:
Department of Mathematics, Faculty of Science Division II, Tokyo University of Science, Tokyo, 162-8601, Japan
*
Corresponding author: Takao Satoh; Email: takao@rs.tus.ac.jp
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Abstract

In this article, we study the Johnson homomorphisms of basis-conjugating automorphism groups of free groups. We construct obstructions for the surjectivity of the Johnson homomorphisms. By using it, we determine their cokernels of degree up to four and give further observations for degree greater than four. As applications, we give the affirmative answer to the Andreadakis problem for the basis-conjugating automorphism groups of free groups at degree four. Moreover, we calculate twisted first cohomology groups of the braid-permutation automorphism groups of free groups.

Keywords

Basis-conjugating automorphism groups of free groupsJohnson homomorphismsAndreadakis–Johnson filtration

MSC classification

Primary: 20F28: Automorphism groups of groups 20F14: Derived series, central series, and generalizations 20C30: Representations of finite symmetric groups
Secondary: 20J06: Cohomology of groups

Information

Type
Research Article
Information
Glasgow Mathematical Journal , First View , pp. 1 - 29
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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