Homogeneous space with non-virtually abelian discontinuous groups but without any proper SL(2, ℝ)-action

Abstract

In the study of discontinuous groups for non-Riemannian homogeneous spaces, the idea of “continuous analogue” gives a powerful method (T. Kobayashi [Math. Ann. 1989]). For example, a semisimple symmetric space [Formula: see text] admits a discontinuous group which is not virtually abelian if and only if [Formula: see text] admits a proper [Formula: see text]-action (T. Okuda [J. Differ. Geom. 2013]). However, the action of discrete subgroups is not always approximated by that of connected groups. In this paper, we show that the theorem cannot be extended to general homogeneous spaces [Formula: see text] of reductive type. We give a counterexample in the case [Formula: see text].