Isometry groups of closed Lorentz 4-manifolds are Jordan
type: Item
instance of: scholarly article
title: Isometry groups of closed Lorentz 4-manifolds are Jordan
zbMATH Open document ID: 1457.57043
published in: Geometriae Dedicata
publication date: 2020-06-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1901.04257
review text: It is known that every Lie group \(G\) with finite number of components has the Jordan property; that is, for some number \(C\) every nontrivial finite subgroup of \(G\) has an abelian proper subgroup of index at most \(C\). In particular, isometry groups of closed Riemannian manifolds are Jordan. However, for closed manifolds with an indefinite metric, the isometry group may have infinitely many components. For manifolds of dimension at most three, the entire diffeomorphism group is known to be Jordan. The author addresses the smallest unknown case and proves that the isometry group of a closed Lorentz 4-manifold is Jordan. The only known examples of closed 4-manifolds whose diffeomorphism group is not Jordan are the two \(S^2\)-bundles over \(T^2\). For these manifolds \(M\) the author proves that, on the other hand, for every positive number \(C\) there exists a Lorentz metric on \(M\) whose automorphism group does not satisfy the Jordan property with this choice of \(C\). So the Jordan property does not hold in a uniform way for these groups.
reviewed by: Rainer Löwen
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 54H15
Mathematics Subject Classification ID: 53C50
Mathematics Subject Classification ID: 20F38
zbMATH DE Number: 7216271
zbMATH Keywords: closed Lorentz manifold
zbMATH Keywords: finite groups of isometries
zbMATH Keywords: Jordan property
author: Ignasi Mundet i Riera
MaRDI profile type: Publication
OpenAlex ID: W2985855524
arXiv ID: 1901.04257
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DOI: 10.1007/S10711-019-00493-7
description: scientific article