On the cohomology of \(\operatorname{SL}_n(\mathbb{Z})\) (Q6592059)

This paper is concerned with the cohomology of the arithmetic group \(\mathrm{SL}_n(\mathbb Z)\), which is of interest to algebraists and number theorists. This group has a classifying space for proper actions of dimension \(\nu_n = \tfrac {n(n-1)}2\) (a retract of its associated locally symmetric space) and in fact its virtual cohomological dimension equals \(\nu_n\). However, it is expected that the groups \(H^{\nu_n - i}(\mathrm{SL}_n(\mathbb Z), \mathbb Q)\) themselves vanish for \(i < (n-1)\) (see [\textit{T. Church} et al., Contemp. Math. 620, 55--70 (2014; Zbl 1377.11065)]). In this paper the author proves that this would be sharp for integers \(n\) congruent to 1 or 0 mod 3 by constructing explicit \(\mathbb Q\)-cocycles in degrees \(\nu_n-i\) for \(i=n-1, n\) which do not vanish in cohomology. The construction is inductive starting from \(n=3, 1, 4\), using a composition of cocycles in the so-called ``sharbly complex'' (introduced a while ago by the authors and collaborators to study this kind of problem) which computes the cohomology via Borel-Serre duality.