Proper action on a homogeneous space of reductive type
type: Item
instance of: scholarly article
title: Proper action on a homogeneous space of reductive type
zbMATH Open document ID: 0662.22008
DOI: 10.1007/BF01443517
author: Toshiyuki Kobayashi
published in: Mathematische Annalen
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164598
review text: An action of L on a homogeneous space G/H is investigated where L,H\(\subset G\) are reductive groups. A criterion of the properness of this action is obtained in terms of the little Weyl group of G. Especially \({\mathbb{R}}\)-rank G\(={\mathbb{R}}\)-rank H iff Calabi-Markus phenomenon occurs, i.e. only finite subgroups can act properly discontinuously on G/H. Then by using cohomological dimension theory of a discrete group, \(L\setminus G/H\) is proved compact iff \(d(G)=d(L)+d(H)\), where d(G) denotes the dimension of Riemannian symmetric space associated with G, etc. These results apply to the existence problem of lattice in G/H. Six series of classical pseudo-Riemannian homogeneous spaces are found to admit non-uniform lattice as well as uniform lattice, while some necessary condition for uniform lattice is obtained when rank G\(=rank H\).
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 43A85
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 4081859
zbMATH Keywords: action
zbMATH Keywords: homogeneous space
zbMATH Keywords: reductive groups
zbMATH Keywords: Weyl group
zbMATH Keywords: cohomological dimension
zbMATH Keywords: Riemannian symmetric space
zbMATH Keywords: pseudo-Riemannian homogeneous spaces
zbMATH Keywords: uniform lattice
reviewed by: Toshihiro Kobayashi
MaRDI profile type: Publication
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OpenAlex ID: W1999500002
description: scientific article; zbMATH DE number 4081859