The action of matrix groups on aspherical manifolds

Shengkui Ye

doi:10.2140/agt.2018.18.2875

The action of matrix groups on aspherical manifolds

Shengkui Ye^*

^*Corresponding author for this work

Department of Pure Mathematics

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

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(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 SL_n(ℤ) cannot act nontrivially on M when r < n. This confirms a conjecture related to Zimmer’s program for these manifolds.

Original language	English
Pages (from-to)	2875-2895
Number of pages	21
Journal	Algebraic and Geometric Topology
Volume	18
Issue number	5
DOIs	https://doi.org/10.2140/agt.2018.18.2875
Publication status	Published - 28 Aug 2018

Keywords

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

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abstract = "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{\textquoteright}s program for these manifolds.",

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AB - 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.

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KW - Matrix group actions

KW - Nil-manifolds

KW - Zimmer’s program

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JF - Algebraic and Geometric Topology

IS - 5

ER -

The action of matrix groups on aspherical manifolds

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Keywords

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