Proper action on a homogeneous space of reductive type. (English) Zbl 0662.22008

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\(=rank H\).
Reviewer: T.Kobayashi


22E40 Discrete subgroups of Lie groups
53C30 Differential geometry of homogeneous manifolds
43A85 Harmonic analysis on homogeneous spaces
53C35 Differential geometry of symmetric spaces
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