An action of L on a homogeneous space G/H is investigated where L,H ⊂ G are reductive Lie groups.
A criterion of the properness of this action is obtained in terms of the little Weyl group of G. In particular, R-rank G = R-rank H iff Calabi-Markus phenomenon occurs, i.e. only finite subgroups of G can act properly discontinuously on G/H. Then by using cohomological dimension theory of a discrete group, L\G/H is proved compact iff d(G) = d(L)+d(H), where d(G) denotes the dimension of a Riemannian symmetric space associated with G, etc.
These results apply to the existence problem of lattice in G/H. Several series of classical pseudo-Riemannian homogeneous spaces are found to admit non-uniform lattice as well as uniform lattice, while some necessary condition for the existence of uniform lattice is obtained when rank G = rank H.
© Toshiyuki Kobayashi