The action of matrix groups on aspherical manifolds

Let ${\mathrm{S}\mathrm{L}}_n(\mathbb{Z})$ $(n\ge 3)$ be the special linear group and $M^r$ be a closed aspherical manifold. It is proved that when $r<n,$ a group action of ${\mathrm{S}\mathrm{L}}_n(\mathbb{Z})$ on $M^r$ by homeomorphisms is trivial if and only if the induced group homomorphism $$\mathrm{S}\mathrm{L}$_{n}(% $\mathbb{Z}$)\rightarrow $\mathrm{O}\mathrm{u}\mathrm{t}$(\pi _{1}(M))$ is trivial. For (almost) flat manifolds, we prove a similar result in terms of holonomy groups. Especially, when ${\pi }_1(M)$ is nilpotent, the group ${\mathrm{S}\mathrm{L}}_n($ cannot act nontrivially on $M$ when $r<n\mathrm{.}$ This confirms a conjecture related to Zimmer’s program for these manifolds.