Algebraic & Geometric Topology

The action of matrix groups on aspherical manifolds

Shengkui Ye

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Let SL n ( ) 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 ( ) on M r by homeomorphisms is trivial if and only if the induced group homomorphism SL n ( ) 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 ( ) cannot act nontrivially on M when r < n . This confirms a conjecture related to Zimmer’s program for these manifolds.

Article information

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

Received: 20 June 2017
Revised: 19 April 2018
Accepted: 3 June 2018
First available in Project Euclid: 30 August 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57S25: Groups acting on specific manifolds 57S20: Noncompact Lie groups of transformations
Secondary: 57S17: Finite transformation groups

Nil-manifolds aspherical manifolds Zimmer's program matrix group actions


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.

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