Vanishing near the virtual cohomological dimension of $\mathrm{SL}_n(\mathbb Z)$

Abstract

We prove the Church--Farb--Putman conjecture that \[ H^{N_n-i}(\mathrm{SL}_n(\mathbb Z);\mathbb Q)=0, \qquad N_n=\binom n2,\quad n\ge2,\quad 0\le i<n-1. \] The proof relies on Franke's filtration of the weighted automorphic cochain complex, developed in his proof of Borel's conjecture. We treat trivial and nontrivial cuspidal support separately. For nontrivial support, we combine a degree calculation with the Chenevier--Lannes lower bound of $11$ for the motivic weight of level-one regular algebraic cuspidal representations of rank at least two. This reduces the contributing inducing data to a trivial factor and a Speh factor attached to such a representation. We construct primitives for closed lifts of cohomology classes in the filtration quotients by combining an Eisenstein residue identity with a cochain homotopy and finite distributions in the spectral parameters. A finite-filtration lifting argument then yields vanishing for nontrivial cuspidal support. The trivial-support case is treated using Franke's topological model.