*Compact Clifford-Klein Forms of Homogeneous Manifolds*,

AMS 1997 Spring Eastern Sectional Meeting, University of
Maryland at College Park, USA, 12-13 April 1997.

Let *G* be a Lie group and *H* its closed subgroup.
We say a discrete subgroup Γ of *G* is a **discontinuous group for** *G*/*H* if the natural action of Γ on *G*/*H* is properly discontinuous.
If the action of Γ is furthermore fixed point free, then double coset space Γ *G*/*H* carries a natural manifold structure, which we say a **Clifford-Klein form** of a homogeneous manifold *G*/*H*.
An important feature in our setting is that *H* is non-compact and that not all discrete subgroup of *G* can act properly discontinuously on *G*/*H*. Fundamental problems are:

*Which homogeneous manifold **G*/*H* admits an infinite discontinuous group?
*Which homogeneous manifold **G*/*H* admits a compact Clifford-Klein form?

Our concern is mainly with reductive cases and we shall present:
- A solution of the so called Calabi-Markus phenomenon.
- A necessary and sufficient condition for discrete subgroups to act properly discontinuously on homogeneous manifolds of reductive groups.
- A sufficient condition for the existence of compact Clifford-Klein forms.
- Conversely, a number of (explicitly computable) obstructions for the existence of compact Clifford-Klein forms of homogeneous manifolds.

© Toshiyuki Kobayashi