Affine groups acting on $\mathbb{R}^{2n+2}$ with the semisimple part $\mathrm{SO}(n+1,n)$

Let \mathrm{Aff}(\mathbb{R}^{d}) be the group of affine transformations of the affine space \mathbb{R}^{d} . Let \Gamma be a subgroup of \mathrm{Aff}(\mathbb{R}^{d}) . In 1991, G. Margulis proved that if the linear part of \Gamma is Zariski dense in \operatorname{SO}(3,2) , then \Gamma does not act properly discontinuously. We then proved that if the linear part of \Gamma is Zariski dense in \operatorname{SO}(n+1,n) , where n is even, then \Gamma does not act properly discontinuously (Abels, Margulis and Soifer, 2011). Based on dynamic ideas, we prove the following theorem, which we announced in Abels, Margulis and Soifer (2020). Theorem. Let \Gamma be an affine group acting on \mathbb{R}^{2n+2} , where n is even. If the semisimple part of the Zariski closure of \Gamma is \operatorname{SO}(n+1,n) , then \Gamma does not act properly discontinuously.