Homogeneous spaces of real simple Lie groups with proper actions of non virtually abelian discrete subgroups: a computational approach. (English) Zbl 1492.22008
Summary: Let \(G\) be a simple non-compact linear connected Lie group and \(H \subset G\) be a closed non-compact semisimple subgroup. We are interested in finding classes of homogeneous spaces \(G / H\) admitting proper actions of discrete non-virtually abelian subgroups \(\Gamma \subset G\). We develop an algorithm for finding such homogeneous spaces. As a testing example we obtain a list of all non-compact homogeneous spaces \(G / H\) admitting proper action of a discrete and non virtually abelian subgroup \(\Gamma\subset G\) in the case when \(G\) has rank at most 8, and \(H\) is a maximal proper semisimple subgroup, provided that the pair \((\mathfrak{g}, \mathfrak{h})\) is contained in a database created by De Graaf and Marrani.
MSC:
| 22E40 | Discrete subgroups of Lie groups |
| 17B20 | Simple, semisimple, reductive (super)algebras |
| 22-08 | Computational methods for problems pertaining to topological groups |
| 22F30 | Homogeneous spaces |
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