#### Volume 18, issue 5 (2018)

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The action of matrix groups on aspherical manifolds

### Shengkui Ye

Algebraic & Geometric Topology 18 (2018) 2875–2895
##### Abstract

Let ${SL}_{n}\left(ℤ\right)$ for $n\ge 3$ be the special linear group and ${M}^{r}$ be a closed aspherical manifold. It is proved that when $r, a group action of ${SL}_{n}\left(ℤ\right)$ on ${M}^{r}$ by homeomorphisms is trivial if and only if the induced group homomorphism ${SL}_{n}\left(ℤ\right)\to Out\left({\pi }_{1}\left(M\right)\right)$ 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 ${SL}_{n}\left(ℤ\right)$ cannot act nontrivially on $M$ when $r. This confirms a conjecture related to Zimmer’s program for these manifolds.

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##### Keywords
Nil-manifolds, aspherical manifolds, Zimmer's program, matrix group actions
##### Mathematical Subject Classification 2010
Primary: 57S25, 57S20
Secondary: 57S17