Discontinuous groups for non-Riemannian homogeneous spaces. (English) Zbl 1023.53031

Engquist, Björn (ed.) et al., Mathematics unlimited - 2001 and beyond. Berlin: Springer. 723-747 (2001).
Let \(G\) be a Lie group and \(H\) a closed subgroup of \(G\). Then the right coset space \(G/H\) can be equipped with a smooth manifold structure so that the canonical projection from \(G\) onto \(G/H\) is a smooth map. Now let \(\Gamma\) be a discrete closed subgroup of \(G\). One says that \(\Gamma\) is a discontinuous group for \(G/H\) if \(\Gamma\) acts properly discontinuously and freely on \(G/H\). If \(\Gamma\) is a discontinuous group for \(G/H\), then the double coset space \(\Gamma \setminus G/H\) carries a natural smooth manifold structure so that the canonical projection from \(G/H\) to \(\Gamma \setminus G/H\) is a local diffeomorphism. The double coset space \(\Gamma \setminus G/H\) is called a Clifford-Klein form of the homogeneous space \(G/H\). In this article the author discusses the following three general problems.
Problem A: Find a criterion for a discrete subgroup \(\Gamma\) to act properly discontinuously on \(G/H\).
Problem B: Determine all possible pairs \((G,H)\) such that \(G/H\) admits a Clifford-Klein form \(\Gamma \setminus G/H\).
Problem C: Describe the moduli of all deformations of a discontinuous group \(\Gamma\) for \(G/H\).
The author provides a motivation for each of these problems and illustrates the difficulties that one encounters when trying to solve them. The current state of affairs is presented, and many examples, open problems and references are given. This article is well-written and gives an excellent introduction to this topic.
For the entire collection see [Zbl 0955.00011].


53C30 Differential geometry of homogeneous manifolds
57S30 Discontinuous groups of transformations
22E40 Discrete subgroups of Lie groups
57S25 Groups acting on specific manifolds
01A67 Future perspectives in mathematics
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry