## 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

### MSC:

 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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### References:

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