## Journal of Differential Geometry

### 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.

#### Article information

Source
J. Differential Geom., Volume 60, Number 2 (2002), 315-344.

Dates
First available in Project Euclid: 20 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1090351104

Digital Object Identifier
doi:10.4310/jdg/1090351104

Mathematical Reviews number (MathSciNet)
MR1938115

Zentralblatt MATH identifier
1061.22012

#### Citation

Abels, H.; Margulis, G.A.; Soifer, G.A. On the Zariski Closure of the Linear Part of a Properly Discontinuous Group of Affine Transformations. J. Differential Geom. 60 (2002), no. 2, 315--344. doi:10.4310/jdg/1090351104. https://projecteuclid.org/euclid.jdg/1090351104