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Discontinuous groups acting on homogeneous spaces of reductive type. (English) Zbl 1193.22010
Kawazoe, T. (ed.) et al., Representation theory of Lie groups and Lie algebras. Proceedings of the conference, Fuji-Kawaguchiko, Japan, August 31–September 3, 1990. Hackensack, NJ: World Scientific (ISBN 981-02-1090-6). 59-75 (1992).
From the text: Let $$H$$ be a closed subgroup of a Lie group $$G$$. The subject of this expository paper is roughly about the following:
Question A-0. How large a discrete subgroup of $$G$$ can act properly discontinuously on a homogeneous space $$G/H$$?
Our concern will be mainly with the case where $$G/H$$ is a homogeneous space of reductive type (Definition 5). If $$H$$ is not compact, the action of a discrete subgroup $$\Gamma$$ of $$G$$ on $$G/H$$ is not automatically properly discontinuous and the double coset $$\Gamma\backslash G/H$$ may be non-Hausdorff. This fact is the main difficulty in our problem. In fact, it may well happen that only finite subgroups of $$G$$ can act properly discontinuously on $$G/H$$. For example, suppose that $$G/H = \text{SO}(n+1,1)/\text{SO}(n, 1)$$, a pseudo-Riemannian homogeneous space of metric type $$(n, 1)$$ and that $$\Gamma$$ is a discrete subgroup of $$G$$. E. Calabi and L. Markus proved that $$\Gamma\backslash G/H$$ is Hausdorff if and only if $$\Gamma$$ is a finite group [Ann. Math. (2) 75, 63–76 (1962; Zbl 0101.21804)]. Thus, a homogeneous space $$G/H = \text{SO}(n+1, 1)/\text{SO}(n, 1)$$ is somehow like a compact space. Named after their surprising discovery, such a homogeneous space is called to have a Calabi-Markus phenomenon.
In contrast to the above case with a noncompact isotropy subgroup $$H$$, A. Borel and Harish-Chandra [Ann. Math. (2) 75, 485–535 (1962; Zbl 0107.14804)] and A. Borel [Proc. Int. Congr. Math., Stockholm 1962, 10–22 (1963; Zbl 0134.16502)] showed that Riemannian symmetric spaces are rich in properly discontinuous actions. That is, let $$G$$ be a real reductive linear Lie group and $$K$$ a maximal compact group of $$G$$. Then there exists a discrete subgroup $$\Gamma$$ of $$G$$ such that the double coset space $$\Gamma\backslash G/H$$ is a compact (Hausdorff smooth) manifold. Also, there exists a discrete subgroup $$\Gamma$$ such that the double coset space is noncompact manifold of finite volume.
On the other hand, even if the isotropy subgroup $$H$$ is noncompact, it may also happen that a homogeneous space has a large discontinuous group $$\Gamma$$ such that $$\Gamma\backslash G/H$$ is a compact manifold. Namely, this is an opposite extremal case to a Calabi-Markus phenomenon. A group manifold $$G/H = G'\times G'/ \text{diag}\,G'$$ is the case. We want to find other homogeneous spaces which admit large discontinuous groups.
For the entire collection see [Zbl 1098.22002].

MSC:
 22E40 Discrete subgroups of Lie groups 22-02 Research exposition (monographs, survey articles) pertaining to topological groups