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

Abstract

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 $\mathrm{SO}(3,2)$, then $\Gamma$ does not act properly discontinuously. We then proved that if the linear part of $\Gamma$ is Zariski dense in $\mathrm{SO}(n+1,n)$, 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 $\mathrm{SO}(n+1,n)$, then $\Gamma$ does not act properly discontinuously.