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On discontinuous subgroups acting on solvable homogeneous spaces. (English) Zbl 1241.22009

This note deals with some features of the deformation space of properly discontinuous actions of a discrete subgroup on a homogeneous space, namely stability and rigidity. A classical local rigidity Theorem (Selberg and Weil, Kobayashi) asserts that there are no continuous deformations of cocompact discontinuous groups for \(G/H\) for a linear noncompact semisimple Lie group \(G\) except for few cases. When \(G\) is nilpotent it is conjectured here that the local rigidity property does not hold. When \(G\) is assumed to be exponential solvable and \(H\subset G\) is a maximal subgroup the author shows that the local rigidity property holds if and only if the group is isomorphic to the group of affine transformations of the line. The author also studied the notion of stable discontinuous subgroups following Kobayashi-Nasrin and he characterized stable discrete subgroups for Heisenberg groups. Unfortunately the detailed proofs cannot be found in this note.

MSC:

22E27 Representations of nilpotent and solvable Lie groups (special orbital integrals, non-type I representations, etc.)
32G05 Deformations of complex structures
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