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On discontinuous groups acting on homogeneous spaces with non-compact isotropy subgroups. (English) Zbl 0815.57029
Summary: Let $$G$$ be a Lie group and $$H$$ a closed subgroup. The action of a discrete subgroup $$\Gamma$$ of $$G$$ on $$G/H$$ is not always properly discontinuous if $$H$$ is noncompact. If the action of $$\Gamma$$ is properly discontinuous, then $$\Gamma$$ is called a discontinuous group acting on $$G/H$$. If $$G/H$$ is of reductive type, it is known that there are no infinite discontinuous groups acting on $$G/H$$ (called Calabi-Markus phenomenon) iff $$\mathbb{R}$$-rank $$G = \mathbb{R}$$-rank $$H$$. For a better understanding of discontinuous groups we are thus interested in cases (i) where $$G/H$$ is nonreductive, and (ii) where $$G/H$$ is of reductive type with $$\mathbb{R}$$-rank $$G = \mathbb{R}$$-rank $$H + 1$$. In this paper we consider the Calabi-Markus phenomenon in solvable cases of type (i). We also study discontinuous groups of reductive group manifolds for case (ii) and generalize a result of Kulkarni-Raymond to higher dimensions.

##### MSC:
 57S30 Discontinuous groups of transformations 57S25 Groups acting on specific manifolds 53C30 Differential geometry of homogeneous manifolds 22E40 Discrete subgroups of Lie groups
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