On the Zariski closure of the linear part of a properly discontinuous group of affine transformations (Q1423998)

Let \(\Gamma\) be a group of affine transformations of \(\mathbb{R}^{2n+1}\) acting properly discontinuously on \(\mathbb{R}^{2n+1}\), and \(\ell(\Gamma)\) denote the subgroup of \(\text{GL}(2n+ 1,\mathbb{R})\) consisting of linear parts of elements of \(\Gamma\). J. Milnor had raised the question (generalising the Auslander conjecture) whether \(\ell(\Gamma)\) is necessarily virtually solvable (viz. admits a subgroup of finite index which is solvable), which was shown by G. A. Margulis to be not the case. In this context there is considerable interest in knowing which (nonsolvable) algebraic subgroups of \(\text{GL}(2n+ 1,\mathbb{R})\) can arise as Zariski closures of subgroups of the form \(\ell(\Gamma)\) as above. In the paper under review it is shown that \(\text{SO}(n+ 1, n)\) contains a Zariski dense subgroup of the form \(\ell(\Gamma)\) as above, if and only if \(n\) is odd.