Church–Farb–Putman parity-based stability conjecture for Aut(F_n) and Out(F_n)

Determine whether, for integers n and i with n sufficiently large relative to i, the rational cohomology groups H^{2n−2−i}(Aut(F_n); Q) and H^{2n−3−i}(Out(F_n); Q) are determined by n+2 (i.e., depend only on the parity of n).

Background

For Aut(F_n) and Out(F_n), the virtual cohomological dimensions are 2n−2 and 2n−3, respectively. Church–Farb–Putman proposed that high-dimensional cohomology in low codimension depends only on the parity of n, rather than vanishing outright.

This conjecture remains open; however, non-vanishing in top degree for Out(F_7) shows any such stabilization can only begin at n ≥ 7, making the conjecture less likely than in the SL_n(Z) and MCG cases.