Clifford-Klein Forms of Non-Riemannian Homogeneous Spaces,
(course lectures), International Conference and Instructional Workshop: Geometry Topology, Analysis of Locally Symmetric Spaces and Discrete Groups (organized by Shing-Tung Yau (chair), Lizhen Ji, Kefeng Liu, Nanhua Xi Hongwei Xu, Lo Yang, Zhu-Jun Zheng and Xiangyu Zhou), Beijing, Republic of China, 17 July-4 August 2006.

In this lecture, I will focus on a new movement on discontinuous groups, that is, discontinuous groups for non-Riemannian spaces.

I will give an up-to-date account of the theory of discrete group actions on non-Riemannian homogeneous spaces.

As an introduction of the motifs, we begin by reviewing the current knowledge of possible global forms of pseudo-Riemannian manifolds with constant curvatures, and discuss what kind of problems we propose to pursue in the first lecture.

For pseudo-Riemannian manifolds, isometric actions of discrete groups are not always properly discontinuous. The fundamental problem is to understand when discrete subgroups of Lie groups G act properly discontinuously on homogeneous spaces G/H for non-compact H. For this, we introduce the concepts from a group-theoretic perspective, including the 'discontinuous dual' of G/H that recovers H in a sense. We then summarize recent results giving criteria for the existence of properly discontinuous subgroups, and the known results and conjectures on the existence of cocompact ones.

We also discuss rigidity/non-rigidity results for cocompact properly discontinuous groups for pseudo-Riemannian symmetric spaces.

Properly discontinuous actions led us to the concept of discretely decomposable restrictions of unitary representations. In the fourth lecture, we plan to give an exposition of the application of the theory of the restriction of unitary representations to modular symbols.

© Toshiyuki Kobayashi