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Selecta Mathematica

, Volume 24, Issue 2, pp 1453–1478 | Cite as

Cohomology of automorphism groups of free groups with twisted coefficients

  • Oscar Randal-Williams
Open Access
Article

Abstract

We compute the groups \(H^*(\mathrm {Aut}(F_n); M)\) and \(H^*(\mathrm {Out}(F_n); M)\) in a stable range, where M is obtained by applying a Schur functor to \(H_\mathbb {Q}\) or \(H^*_\mathbb {Q}\), respectively the first rational homology and cohomology of \(F_n\). The answer may be described in terms of stable multiplicities of irreducibles in the plethysm \(\mathrm {Sym}^k \circ \mathrm {Sym}^l\) of symmetric powers. We also compute the stable integral cohomology groups of \(\mathrm {Aut}(F_n)\) with coefficients in H or \(H^*\).

Keywords

Automorphisms of free groups Homology stability 

Mathematics Subject Classification

20F28 20J06 57R20 

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Centre for Mathematical SciencesCambridgeUK

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