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.


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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