## T. Kobayashi,

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

Ph.D. thesis, the University of Tokyo, 1990.
DOI: 10.11501/3087251.

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*.

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© Toshiyuki Kobayashi