## Algebraic & Geometric Topology

### The action of matrix groups on aspherical manifolds

Shengkui Ye

#### Abstract

Let $SL n ( ℤ )$ for $n ≥ 3$ be the special linear group and $M r$ be a closed aspherical manifold. It is proved that when $r < n$, a group action of $SL n ( ℤ )$ on $M r$ by homeomorphisms is trivial if and only if the induced group homomorphism $SL n ( ℤ ) → Out ( π 1 ( M ) )$ is trivial. For (almost) flat manifolds, we prove a similar result in terms of holonomy groups. In particular, when $π 1 ( M )$ is nilpotent, the group $SL n ( ℤ )$ cannot act nontrivially on $M$ when $r < n$. This confirms a conjecture related to Zimmer’s program for these manifolds.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 5 (2018), 2875-2895.

Dates
Revised: 19 April 2018
Accepted: 3 June 2018
First available in Project Euclid: 30 August 2018

https://projecteuclid.org/euclid.agt/1535594425

Digital Object Identifier
doi:10.2140/agt.2018.18.2875

Mathematical Reviews number (MathSciNet)
MR3848402

Zentralblatt MATH identifier
06935823

#### Citation

Ye, Shengkui. The action of matrix groups on aspherical manifolds. Algebr. Geom. Topol. 18 (2018), no. 5, 2875--2895. doi:10.2140/agt.2018.18.2875. https://projecteuclid.org/euclid.agt/1535594425

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