Selecta Mathematica

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

Cohomology of automorphism groups of free groups with twisted coefficients

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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Authors and Affiliations

  1. 1.Centre for Mathematical SciencesCambridgeUK

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