On discontinuous groups acting on homogeneous spaces with non-compact isotropy subgroups

Abstract

Let G be a Lie group and H a closed subgroup. The action of a discrete subgroup ɼ of G on G/ H is not always properly discontinuous if H is non-compact. If the action of ɼ is properly discontinuous, then ɼ 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 R -rank G = R-rank H . For a better understanding of discontinuous groups we are thus interested in cases (i) where G/ H is non-reductive, and (ii) where G/ H is of reductive type with R -rank G = 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.