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


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 \(H^*\).


Automorphisms of free groups Homology stability 

Mathematics Subject Classification

20F28 20J06 57R20 


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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, 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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