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].

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].