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Let $\mathrm{SL}_{n}(\mathbb{Z})$ $(n\geq 3)$ be the special linear group and$M^{r}$ be a closed aspherical manifold. It is proved that when $r<n,$ a groupaction of $\mathrm{SL}_{n}(\mathbb{Z})$ on $M^{r}$ by homeomorphisms is trivialif and only if the induced group homomorphism $\mathrm{SL}_{n}(%\mathbb{Z})\rightarrow \mathrm{Out}(\pi _{1}(M))$ is trivial. For (almost) flatmanifolds, we prove a similar result in terms of holonomy groups. Especially,when $\pi _{1}(M)$ is nilpotent, the group $\mathrm{SL}_{n}(% \mathbb{Z})$cannot act nontrivially on $M$ when $r<n.$ This confirms a conjecture relatedto Zimmer's program for these manifolds.

The action of matrix groups on aspherical manifolds