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 \(H^*\).
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Supported by ERC Advanced Grant No. 228082, the Danish National Research Foundation through the Centre for Symmetry and Deformation, and EPSRC Grant EP/M027783/1.
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Randal-Williams, O. Cohomology of automorphism groups of free groups with twisted coefficients. Sel. Math. New Ser. 24, 1453–1478 (2018). https://doi.org/10.1007/s00029-017-0311-0
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DOI: https://doi.org/10.1007/s00029-017-0311-0