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On the low-dimensional cohomology groups of the IA-automorphism group of the free group of rank three
Part of:
Special aspects of infinite or finite groups
Connections with homological algebra and category theory
Published online by Cambridge University Press: 05 May 2021
Abstract
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In this paper, we study the structure of the rational cohomology groups of the IA-automorphism group $\mathrm {IA}_3$
of the free group of rank three by using combinatorial group theory and representation theory. In particular, we detect a nontrivial irreducible component in the second cohomology group of $\mathrm {IA}_3$
, which is not contained in the image of the cup product map of the first cohomology groups. We also show that the triple cup product of the first cohomology groups is trivial. As a corollary, we obtain that the fourth term of the lower central series of $\mathrm {IA}_3$
has finite index in that of the Andreadakis–Johnson filtration of $\mathrm {IA}_3$
.
Keywords
MSC classification
Primary:
20F28: Automorphism groups of groups
Secondary:
20J06: Cohomology of groups
- Type
- Research Article
- Information
- Creative Commons
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.- Copyright
- Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society
Footnotes
Dedicated to Professor Shigeyuki Morita on the occasion of his 70th birthday.
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