On the Zariski Closure of the Linear Part of a Properly Discontinuous Group of Affine Transformations

Abstract

Let $\Gamma$ be a subgroup of the group of affine transformations of the affine space $\mathbb{R}^{2n+1}$. Suppose $\Gamma$ acts properly discontinuously on $\mathbb{R}^{2n+1}$. The paper deals with the question which subgroups of $\rm{GL}(2n +1,\mathbb{R})$ occur as Zariski closure $\overline{\ell (\Gamma})$ of the linear part of such a group $\Gamma$. The two main results of the paper say that $\rm{SO}(n + 1, n)$ does occur as $\overline{\ell (\Gamma)}$ of such a group$\Gamma$ if $n$ is odd, but does not if $n$ is even.