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

Access to Document

10.2140/agt.2018.18.2875

Cite this

@article{b0da2bedd832427d878d7995e6ddb097,

title = "The action of matrix groups on aspherical manifolds",

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.",

keywords = "Aspherical manifolds, Matrix group actions, Nil-manifolds, Zimmer{\textquoteright}s program",

author = "Shengkui Ye",

year = "2018",

month = aug,

day = "28",

doi = "10.2140/agt.2018.18.2875",

language = "English",

volume = "18",

pages = "2875--2895",

journal = "Algebraic and Geometric Topology",

issn = "1472-2747",

number = "5",

}

TY - JOUR

T1 - The action of matrix groups on aspherical manifolds

AU - Ye, Shengkui

PY - 2018/8/28

Y1 - 2018/8/28

N2 - 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.

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.

KW - Aspherical manifolds

KW - Matrix group actions

KW - Nil-manifolds

KW - Zimmer’s program

UR - http://www.scopus.com/inward/record.url?scp=85052652678&partnerID=8YFLogxK

U2 - 10.2140/agt.2018.18.2875

DO - 10.2140/agt.2018.18.2875

M3 - Article

AN - SCOPUS:85052652678

SN - 1472-2747

VL - 18

SP - 2875

EP - 2895

JO - Algebraic and Geometric Topology

JF - Algebraic and Geometric Topology

IS - 5

ER -

The action of matrix groups on aspherical manifolds

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this