The action of matrix groups on aspherical manifolds

Shengkui Ye*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Citation (Scopus)


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

Original languageEnglish
Pages (from-to)2875-2895
Number of pages21
JournalAlgebraic and Geometric Topology
Issue number5
Publication statusPublished - 28 Aug 2018


  • Aspherical manifolds
  • Matrix group actions
  • Nil-manifolds
  • Zimmer’s program


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