Some remarks on area-preserving actions of lattices
type: Item
instance of: scholarly article
zbMATH Open document ID: 1260.22015
author: Pierre Py
publication date: 2012-09-05T00:00:00Z
Mathematics Subject Classification ID: 22F05
Mathematics Subject Classification ID: 37A10
Mathematics Subject Classification ID: 22A26
Mathematics Subject Classification ID: 37C85
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 6077147
zbMATH Keywords: lattices
zbMATH Keywords: Lie group
zbMATH Keywords: diffeomorphism
zbMATH Keywords: spectrum
MaRDI profile type: Publication
title: Some remarks on area-preserving actions of lattices
review text: The existence of area-preserving actions of higher-rank lattices on surfaces is discussed. A lattice \(\Gamma\) in a connected simple Lie group \(G\) with finite center and of real rank greater than \(1\) is considered. Let \(\Sigma\) be a closed oriented surface endowed with an area form \(\omega\). According to a conjecture of Zimmer any homomorphism \(\rho:\Gamma \rightarrow \operatorname{Dif}f (\Sigma ,\omega )\) from \(\Gamma\) to the group of area-preserving diffeomorphisms of \(\Sigma\) should have finite image. A consequence of Zimmer's results is that the group \(\rho (\Gamma )\) does not contain any area-preserving diffeomorphism with positive metric entropy for some measure \(\mu\). On the other side of the spectrum of possible dynamical behaviours are area-preserving diffeomorphisms that are contained in a Hamiltonian flow. A consequence of this discussion is that this kind of diffeomorphism cannot appear in the group \(\rho (\Gamma)\) (see Section 3 of the paper).NEWLINENEWLINEFor the entire collection see [Zbl 1225.00042].
reviewed by: Victor Sharapov
description: scientific article; zbMATH DE number 6077147