Church–Farb–Putman vanishing conjecture for SL_n(Z)

Establish that for integers n and i with n ≥ i+2, the rational cohomology group H^{binomial(n,2)−i}(SL_n(Z); Q) vanishes.

Background

The group SL_n(Z) has virtual cohomological dimension vcd(SL_n(Z)) = binomial(n,2). Church–Farb–Putman formulated a vanishing pattern for its high-dimensional rational cohomology, equivalent to a stability statement via Borel–Serre duality. This conjecture predicts vanishing of cohomology in low codimension (near the top degree).

Progress includes proofs in codimensions i = 0, 1, 2, but the full conjecture remains open. Known non-vanishing phenomena close to the top degree indicate the conjecture (if true) is sharp or nearly sharp.