On discontinuous group actions on non-Riemannian homogeneous spaces

Abstract

This article gives an up-to-date account of the theory of discrete group actions on non-Riemannian homogeneous spaces. As an introduction of the motifs of this article, 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. 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. The final section discusses the deformation theory and in particular rigidity results for cocompact properly discontinuous groups for pseudo-Riemannian symmetric spaces.