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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 G/H admits adiscontinuous group which is not virtually abelian if and only if G/H admits aproper SL(2,R)-action (T. Okuda [J. Differential Geom. 2013]). However, theaction of discrete subgroups is not always approximated by that of connectedgroups. In this paper, we show that the theorem cannot be extended to generalhomogeneous spaces G/H of reductive type. We give a counterexample in the caseG = SL(5,R).

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