The action of matrix groups on aspherical manifolds

TL;DR

This paper investigates actions of the special linear group on aspherical manifolds, establishing conditions under which such actions are trivial, especially for flat and nilpotent cases, supporting conjectures in Zimmer's program.

Contribution

It proves that for certain aspherical manifolds, group actions of SL(n,Z) are trivial when the manifold's dimension is less than n, confirming a conjecture in Zimmer's program.

Findings

SL(n,Z) acts trivially on aspherical manifolds when r<n.

For flat manifolds, triviality depends on holonomy groups.

Nilpotent fundamental groups prevent nontrivial actions when r<n.

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.