The action of matrix groups on aspherical manifolds

Abstract

Let ${\mathrm{SL}}_n(\mathbb{Z})$ $(n\ge 3)$ be the special linear group and $M^r$ be a closed aspherical manifold. It is proved that when $r<n,$ a group action of ${\mathrm{SL}}_n(\mathbb{Z})$ on $M^r$ by homeomorphisms is trivial if and only if the induced group homomorphism ${\mathrm{SL}}_n($ 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 ${\mathrm{SL}}_n($ cannot act nontrivially on $M$ when $r<n.$ This confirms a conjecture related to Zimmer's program for these manifolds.

The article arXiv:1609.07699 is split into two parts. This is the second part. The title is also changed. To appear in Algebraic and Geometric Topology

References in corpus (1)

• Outer actions of $\mathrm{Out}(F_n)$ on small right-angled Artin groups

Cited by in corpus (1)

• Symmetries of flat manifolds, Jordan property and the general Zimmer program