Homogeneous space with non-virtually abelian discontinuous groups but without any proper \(SL(2,\mathbb{R})\)-action (Q2797902)

In previous work [J. Differ. Geom. 94, No. 2, 301--342 (2013; Zbl 1312.53076)], the author classified pseudo-Riemannian semisimple symmetric spaces \(G/H\) that admit a proper action of a three-dimensional simple subgroup of \(G\). Moreover, he showed that the existence of such an action is equivalent to the existence of a properly discontinuous action of a discrete, non-virtually abelian subgroup of \(G\).NEWLINENEWLINEIn the paper at hand the author shows that this theorem is not true without the assumption of \(G/H\) being symmetric, by giving an explicit example of a homogeneous space of \(\mathrm{SL}(5,\mathbb R)\) with a properly discontinuous action of a discrete, non-virtually abelian subgroup, but without any proper action of a three-dimensional simple subgroup of \(G\).