On the zariski closure of the linear part of a properly discontinuous group of affine transformations

Let Γ be a subgroup of the group of affine transformations of the affine space ℝ²ⁿ⁺¹. Suppose Γ acts properly discontinuously on ℝ²ⁿ⁺¹. The paper deals with the question which subgroups of GL(2n+1,ℝ) occur as Zariski closure ℓ(Γ)^¯ of the linear part of such a group Γ. The two main results of the paper say that SO(n + 1, n) does occur as ℓ(Γ)^¯ of such a group Γ if n is odd, but does not if n is even.