On the codimension-two cohomology of \(\mathrm{SL}_{n}(\mathbb{Z})\) (Q6592209)

Using an explicit partial projective resolution of the Steinberg module, the authors prove the codimension 2 part of a conjecture of Church-Farb-Putman on the vanishing of the \(\operatorname{SL}_2(\mathbb{Z})\) cohomology.\N\NIn detail: the Steinberg-module \(St\) attached to the group \(G=\mathrm{SL}_n(\mathbb{Q})\), \(n\ge 3\) is the top homology group \(H_{n-2}(T_n)\), where \(T_n\) is the Tits building of \(G\). It becomes a \(G\)-module through the action of \(G\) on the building. By the work of \textit{A. Borel} and \textit{J.-P. Serre} [Topology 15, 211--232 (1976; Zbl 0338.20055)], it is a dualising module in the sense that for every finite index subgroup \(\Gamma\subset\operatorname{SL}_2(\mathrm{Z})\) one has \(H^k(\Gamma,\mathbb{Q})=0\) for \(k>k_0=\frac{n(n-1}2\) and \N\[\N H^{k_0-i}(\Gamma,\mathbb{Q})\cong H_i(\Gamma,St\otimes\mathbb{Q}). \N\]\NThe main result of the paper is a construction of an explicit projective resolution of the Steinberg module up to the second step, which allows them to deduce, that for instance \(H^{\frac{n(n-1}2-2}(\operatorname{SL}_2(\mathbb{Z}),\mathbb{Q})=0\) for \(n\ge 3\). This resolves the codimension 2 part of the Church-Farb-Puman Conjecture [\textit{T. Church} et al., Contemp. Math. 620, 55--70 (2014; Zbl 1377.11065)].\N\NIn the other direction, for the congruence subgroups \(\Gamma_n(p)\) for \(p=3\) or \(p=5\) the authors deduce that \N\[\N\dim\ H^{\frac{n(n-1)}2-1}(\Gamma_n(p),\mathbb{Q})\ \ge\ p^{\frac{(n-2)(n-3)}2}\ |Gr_2^n(\mathbb{F}_p|\ \left(\frac{p-1}2\right)^{n-2},\N\]\Nwhere \(GR_2^n\) denotes the Grassmannian of \(2\)-planes in \(\mathbb{F}_p^n\).