The action of matrix groups on aspherical manifolds

Abstract

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 group action of $\mathrm{SL}_{n}(\mathbb{Z})$ on $M^{r}$ by homeomorphisms is trivial if and only if the induced group homomorphism $\mathrm{SL}_{n}(% \mathbb{Z})\rightarrow \mathrm{Out}(\pi _{1}(M))$ 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 $\mathrm{SL}_{n}(% \mathbb{Z})$ cannot act nontrivially on $M$ when $r<n.$ This confirms a conjecture related to Zimmer's program for these manifolds.