The action of matrix groups on aspherical manifolds

Abstract

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