(2 lectures), The 2000 Twente Conference on Lie Groups, University of Twente, Enschede, the Netherlands, 18-20 December 2000.

LetGbe a Lie group andHits closed subgroup. We say a discrete subgroup Γ ofGis adiscontinuous group forG/Hif the natural action of Γ onG/His properly discontinuous. If the action of Γ is furthermore fixed point free, then the double coset space Γ\G/Hcarries a natural manifold structure, which we say aClifford-Klein formof a homogeneous manifoldG/H. An important feature in our setting is thatHis non-compact and that not all discrete subgroup ofGcan act properly discontinuously onG/H. Fundamental problems are:We shall discuss the above problems, especially in the case where (

When does a discrete subgroup ofGact properly discontinuously onG/H?Which homogeneous manifoldG/Hadmits an infinite discontinuous group?Which homogeneous manifoldG/Hadmits a compact Clifford--Klein form?Describe deformations of a Clifford-Klein form ofG/H.G, H) is a pair of reductive Lie groups so thatG/Hcarries naturally a pseudo-Riemannian metric.

References

[1] T. Kobayashi, ''Discontinuous groups for non-Riemannian homogeneous spaces'', inMathematics Unlimited, 2001 and Beyond, B. Engquist and W. Schmid, eds., Springer-Verlag, in press.

© Toshiyuki Kobayashi