Summer School of mathematics 2022By a result of Church-Putman, the rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$ vanishes in "codimension one", i.e. $H^{{n \choose 2} -1}(\operatorname{SL}_n(\mathbb{Z});\mathbb{Q}) = 0$ for $n \geq 3$, where ${n \choose 2}$ is the virtual cohomological dimension of $\operatorname{SL}_n(\mathbb{Z})$. I will talk about work in progress on two generalisations of this result: The first project is joint work with Miller-Patzt-Sroka-Wilson (see https://arxiv.org/abs/2204.11967). We show that the rational cohomology of $\operatorname{SL}_n(\mathbb{Z})$ vanishes in codimension two, i.e. $H^{{n \choose 2} -2}(\operatorname{SL}_n(\mathbb{Z});\mathbb{Q}) = 0$ for $n \geq 3$. The second project is joint with Patzt-Sroka. Its aim is to study whether the rational cohomology of the symplectic group $\operatorname{Sp}_{2n}(\mathbb{Z})$ vanishes in codimension one, i.e. whether $H^{n^2 -1}(\operatorname{Sp}_{2n}(\mathbb{Z});\mathbb{Q}) = 0$ for $n \geq 2$.