Discontinuous Groups for Non-Riemannian Homogeneous Manifolds and Deformations of Clifford-Klein Forms,
(2 lectures), The 2000 Twente Conference on Lie Groups, University of Twente, Enschede, the Netherlands, 18-20 December 2000.

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 the 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:

1. When does a discrete subgroup of G act properly discontinuously on G/H?
2. Which homogeneous manifold G/H admits an infinite discontinuous group?
3. Which homogeneous manifold G/H admits a compact Clifford--Klein form?
4. Describe deformations of a Clifford-Klein form of G/H.

We shall discuss the above problems, especially in the case where (G, H) is a pair of reductive Lie groups so that G/H carries naturally a pseudo-Riemannian metric.

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

© Toshiyuki Kobayashi