On Lipsman's property and related questions

In my talk,I will describe approaches to "Lipsman's property", which is considered as a "continuous analogue" of the famous (and still unsolved) Auslander's conjecture. I will consider two properties of the action of a subgroup L of a Lie group G on a homogeneous space G/H, that is, contact intersection property (CI) and properness. The general problem is: what structural implications are forced on L by properness or (CI)? In particular, we say that a homogeneous space G/H has Lipsman's property, if for any Lie subgroup L there is an equivalence between properness and (CI). Lipsman showed that in "generic cases" one needs to prove such equivalence only for solvable G. We propose a new approach to the problem based on our analysis of the behavior of the above equivalence under extensions and using De Graaf algorithms of extending representations of solvable Lie algebras.