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

Let SLn() for n 3 be the special linear group and Mr be a closed aspherical manifold. It is proved that when r < n, a group action of SLn() on Mr by homeomorphisms is trivial if and only if the induced group homomorphism SLn() Out(π1(M)) is trivial. For (almost) flat manifolds, we prove a similar result in terms of holonomy groups. In particular, when π1(M) is nilpotent, the group SLn() cannot act nontrivially on M when r < n. This confirms a conjecture related to Zimmer’s program for these manifolds.

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Nil-manifolds, aspherical manifolds, Zimmer's program, matrix group actions
Mathematical Subject Classification 2010
Primary: 57S25, 57S20
Secondary: 57S17
Received: 20 June 2017
Revised: 19 April 2018
Accepted: 3 June 2018
Published: 22 August 2018
Shengkui Ye
Department of Mathematical Sciences
Xi’an Jiaotong-Liverpool University