T. Kobayashi,

*Proper action on a homogeneous space of reductive type*,

Math. Ann.**285** (1989), no. 2, 249-263..

Math. Ann.

An action of[ full text(pdf) | SpringerLink | GDZ | ZMath | Ph.D. thesis | related papers ]Lon a homogeneous spaceG/His investigated whereL,H⊂Gare 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-rankG=R-rankHiff Calabi-Markus phenomenon occurs, i.e. only finite subgroups ofGcan act properly discontinuously onG/H. Then by using cohomological dimension theory of a discrete group,L\G/His proved compact iffd(G) =d(L)+d(H), whered(G) denotes the dimension of a Riemannian symmetric space associated withG, 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 rankG= rankH.

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© Toshiyuki Kobayashi