Proper actions on strongly regular homogeneous spaces (Q1704287)

Let \(G\) be a connected linear reductive real Lie group and \(H \subset G\) be a closed Lie subgroup of \(G\) with finitely many connected components. Proper actions of some subgroups of \(G\) on the homogeneous space \(G/H\) are investigated. A notion of strongly regular homogeneous spaces is introduced. This notion is a real analogue of the notion of regular complex homogeneous spaces. As a main result the equivalence between two conditions C2 (the space \(G/H\) admits a properly discontinuous action of a non-virtually abelian infinite subgroup of \(G\)) and C3 (the space \(G/H\) admits a proper action of a subgroup locally isomorphic to \(\mathrm{SL}(2, \mathbf R)\)) with some condition in terms of the hyperbolic rank is proved.