Introduction to actions of discrete groups on pseudo-Riemannian homogeneous manifolds
type: Item
instance of: scholarly article
title: Introduction to actions of discrete groups on pseudo-Riemannian homogeneous manifolds
zbMATH Open document ID: 1026.57029
DOI: 10.1023/A:1019778805173
author: Toshiyuki Kobayashi
published in: Acta Applicandae Mathematicae
publication date: 2003-03-11T00:00:00Z
review text: This article is based on lectures delivered by the author at the 2000 Twente Conference on Lie groups. Their goal was ``to give an accessible exposition by clarifying the current status of some of the central problems of this topic''. The following two problems for homogeneous spaces \(G/H\), where \(G\) and \(H\) are reductive real linear noncompact groups, are considered: A) find a criterion for a discrete subgroup \(\Gamma\subset G\) to act properly discontinuously on \(G/H\); B) determine pairs \(G,H\) such that \(G/H\) admits a subgroup \(\Gamma\subset G\) whose action on \(G/H\) is cocompact, properly discontinuous, and free (in this case, \(\Gamma\backslash G/H\) is said to be a Clifford-Klein form for \(G/H\)). Solutions to A) were independently given by the author and Benoist. Any real reductive linear group admits the Cartan decomposition \(G=KAK\); the Cartan projection \(\nu\) is defined by \(\nu(g)=\log a\pmod W\) for \(g=k_1ak_2\), where \(W\) is the Weyl group of \(G\). The mapping \(\nu\) keeps two special relations between subsets \(L,H\) of groups: a) \(L\mathbin\cap SHS\) is relatively compact for any compact set \(S\); b) there exists a compact \(S\) such that \(L\subseteq SHS\) and \(H\subseteq SLS\). If \(L\) and \(H\) are closed subgroups of \(G\) then \(L\) acts properly on \(G/H\) if and only if they satisfy a). On the other hand, it is easy to work with these relations in the abelian group \(A\). Also, there is a kind of duality between a) and b). In particular, these facts yield the Calabi-Marcus phenomenon: the absence of infinite discontinuous groups on \(G/H\) is equivalent to the condition that real ranks of \(G\) and \(H\) are equal. For B), there is no complete solution. If \(H\) is compact then a compact Clifford-Klein form for \(G/H\) exists if and only if \(G\) contains a cocompact discrete subgroup \(\Gamma\); according to Borel's theorem, this is true for all real reductive linear groups (moreover, \(\Gamma\) can be assumed to b…
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 22F30
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 1879587
zbMATH Keywords: discontinuous group
zbMATH Keywords: pseudo-Riemannian manifold
zbMATH Keywords: homogeneous space
zbMATH Keywords: Clifford-Klein form
zbMATH Keywords: discrete subgroup
zbMATH Keywords: Calabi-Marcus phenomenon
reviewed by: Victor Gichev
MaRDI profile type: Publication
full work available at URL: a:1019778805173
OpenAlex ID: W1504942632
description: scientific article; zbMATH DE number 1879587