Affine groups acting on R²ⁿ⁺² with the semisimple part SO(n+1,n)

Let Aff(Rᵈ) be the group of affine transformations of the affine space Rᵈ . Let Γ be a subgroup of Aff(Rᵈ) . In 1991, G. Margulis proved that if the linear part of Γ is Zariski dense in SO(3,2) , then Γ does not act properly discontinuously. We then proved that if the linear part of Γ is Zariski dense in SO(n+1,n) , where n is even, then Γ 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 Γ be an affine group acting on R²ⁿ⁺² , where n is even. If the semisimple part of the Zariski closure of Γ is SO(n+1,n) , then Γ does not act properly discontinuously.