The action of matrix groups on aspherical manifolds

Ye, Shengkui

doi:10.2140/agt.2018.18.2875

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the Journal

Editorial Board

Subscriptions

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1472-2739 (online)

ISSN 1472-2747 (print)

Author Index

To Appear

Other MSP Journals

The action of matrix groups on aspherical manifolds

Shengkui Ye

Algebraic & Geometric Topology 18 (2018) 2875–2895

DOI: 10.2140/agt.2018.18.2875

Abstract

Let ${SL}_{n} (ℤ)$ for $n \geq 3$ 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 ${SL}_{n} (ℤ) \to Out (π_{1} (M))$ 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 ${SL}_{n} (ℤ)$ cannot act nontrivially on $M$ when $r < n$ . This confirms a conjecture related to Zimmer’s program for these manifolds.

Keywords

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

Mathematical Subject Classification 2010

Primary: 57S25, 57S20

Secondary: 57S17

References

Publication

Received: 20 June 2017

Revised: 19 April 2018

Accepted: 3 June 2018

Published: 22 August 2018

Authors

Shengkui Ye
Department of Mathematical Sciences Xi’an Jiaotong-Liverpool University Jiangsu China
https://yeshengkui.wordpress.com/

Volume 18, issue 5 (2018)